An elliptic curve \(E\) over a field \(k\) is a smooth projective curve of genus \(1\) together with a distinguished \(k\)-rational point \(O\).

The most commonly used model for elliptic curves is a Weierstrass model: a smooth plane cubic with the point \(O\) as the unique point at infinity.

An elliptic curve \(E\) defined over a number field \(K\) is said to have additive reduction at a prime \(\mathfrak {p}\) of \(K\) if the reduction of \(E\) modulo \(\mathfrak {p}\) has a cuspidal singularity.

The Tate-Shafarevich group \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \) of an elliptic curve \(E\) defined over a number field \(K\) is a torsion abelian group, which can be defined in terms of Galois cohomology as

where \(v\) runs over all places of \(K\) (finite and infinite), \(K_v\) is the completion of \(K\) at \(v\), \(E_{K_v}\) is the base change of \(E\) to \(K_v\), and \(G_K\) and \(G_{K_v}\) denote absolute Galois groups.

The group \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \) is conjectured to be finite, and its order appears in the strong form of the Birch-Swinnerton-Dyer Conjecture for \(E\). The order implied by the conjecture is called the analytic order of Sha and can be defined as the real number

Here \(D_K\) is the discriminant of \(K\), \(L(E,s)\) is the \(L\)-function of \(E/K\), \(r\) is the analytic rank of \(E/K\), \(\mathrm{Reg}_{\rm NT}(E/K)\) is the Néron-Tate (un-normalised) regulator of \(E/K\), \(E(K)_{\rm tor}\) is the torsion subgroup of the Mordell-Weil group \(E(K)\), \(\Omega (E/K)\) is the global period of \(E/K\), and \(c_{\mathfrak {p}}\) is the Tamagawa number of \(E\) at the prime \(\mathfrak {p}\) of \(K\).

It is known that if \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \) is finite then its order is a square, so one expects the real number \(\mathop{\mathrm{\unicode {x0428}}}\nolimits _{\text{an}}\) to always be a square integer.

For elliptic curves defined over \(\mathbb {Q}\) of rank \(0\) or \(1\), it is a theorem that \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \)_an is a positive rational number, and this rational number can in principle be computed exactly. This exact computation has only been carried out for the curves in the database with rank \(0\). For curves of rank \(2\) and above, there is no such theorem, and the values computed are floating point approximate values which are very close to integers. In the LMFDB we store and display the rounded values in this case.

An elliptic curve \(E\) defined over a number field \(K\) is said to have bad reduction at a prime \(\mathfrak {p}\) of \(K\) if the reduction of \(E\) modulo \(\mathfrak {p}\) is singular. There are three types of bad reduction:

• split multiplicative,

• non-split multiplicative,

• additive.

A curve has bad reduction at \(\mathfrak {p}\) if and only if \(\mathfrak {p}\) divides its discriminant.

If \(E\) is an elliptic curve defined over a field \(K\), and \(L\) is an extension field of \(K\), then the same equation defining \(E\) as an elliptic curve over \(K\) also defines a curve over \(L\) called the base change of \(E\) from \(K\) to \(L\). Any curve defined over \(L\) which is isomorphic to \(E\) over \(L\) is called a base-change curve from \(K\) to \(L\). A sufficient but not necessary condition for a curve to be a base change is that the coefficients of its Weierstrass equation lie in \(K\).

When \(K=\mathbb {Q}\) and \(L\) is a number field, elliptic curves over \(L\) which are base-changes of curves over \(\mathbb {Q}\) may simply be called base-change curves. A necessary, but not sufficient, condition for this is that the \(j\)-invariant of \(E\) should be in \(\mathbb {Q}\).

The Birch and Swinnerton-Dyer conjecture (BSD) is one of the Millennium Prize Problems listed by the Clay Mathematics Institute. It relates the order of vanishing (or analytic rank) and the leading coefficient of the L-function associated to an elliptic curve \(E\) defined over a number field \(K\) at the central point \(s=1\) to certain arithmetic data, the BSD invariants of \(E\).

• The weak form of the BSD conjecture states just that the analytic rank \(r_{an}\) (that is, the order of vanishing of vanishing of \(L(E,s)\) at \(s=1\)), is equal to the rank \(r\) of \(E/K\).

• The strong form of the conjecture states that \(r=r_{an}\) and also that the leading coefficient of the L-function is given by the formula

The quantities appearing in this formula are as follows:

• \(d_K\) is the discriminant of \(K\);

• \(r\) is the rank of \(E(K)\);

• \(\mathop{\mathrm{\unicode {x0428}}}\nolimits (E/K)\) is the Tate-Shafarevich group

of \(E/K\);

• \(\mathrm{Reg}(E/K)\) is the regulator of \(E/K\);

• \(\Omega (E/K)\) is the global period of \(E/K\);

• \(c_{\mathfrak {p}}\) is the Tamagawa number of \(E\) at each prime \(\mathfrak {p}\) of \(K\);

• \(E(K)_{\rm tor}\) is the torsion subgroup of \(E(K)\).

Implicit in the strong form of the conjecture is that the Tate-Shafarevich group \(\mathop{\mathrm{\unicode {x0428}}}\nolimits (E/K)\) is finite.

There is a similar conjecture for abelian varieties over number fields.

Let \(E\) be an elliptic curve defined over a number field \(K\). The canonical height on \(E\) is a function

defined on the Mordell-Weil group \(E(K)\) which induces a positive definite quadratic form on \(E(K)\otimes \mathbb {R}\).

One definition of \(\hat{h}(P)\) is

where \(h(x)\) is the Weil height of \(x\in K\). This definition gives the non-normalised height. A normalised height which is invariant under base-change is given by

Related to the canonical height is the height pairing

defined by \(\langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))\), which is a positive definite quadratic form on \(E(K)\otimes \mathbb {R}\), used in defining the regulator of \(E/K\).

An elliptic curve whose endomorphism ring is larger than \(\mathbb {Z}\) is said to have complex multiplication (often abbreviated to CM). In this case, for curves defined over fields of characteristic zero, the endomorphism ring is isomorphic to an order in an imaginary quadratic field. The discriminant of this order is the CM discriminant.

An elliptic curve whose geometric endomorphism ring is larger than \(\mathbb {Z}\) is said to have potential complex multiplication (potential CM). In the literature, these too are often called CM elliptic curves.

The property of having potential CM depends only on the \(j\)-invariant of the curve. In characteristic \(0\), CM \(j\)-invariants are algebraic integers, and there are only finitely many in any given number field. There are precisely 13 CM \(j\)-invariants in \(\mathbb {Q}\) (all integers), associated to the 13 imaginary quadratic orders of class number \(1\):

CM elliptic curves are examples of CM abelian varieties.

The conductor of an elliptic curve \(E\) defined over a number field \(K\) is an ideal of the ring of integers of \(K\) that is divisible by the prime ideals of bad reduction and no others. It is defined as

where the exponent \(e_{\mathfrak {p}}\) is as follows:

• \(e_{\mathfrak {p}}=0\) if \(E\) has good reduction at \(\mathfrak {p}\);

• \(e_{\mathfrak {p}}=1\) if \(E\) has multiplicative reduction at \(\mathfrak {p}\);

• \(e_{\mathfrak {p}}=2\) if \(E\) has additive reduction at \(\mathfrak {p}\) and \(\mathfrak {p}\) does not lie above either \(2\) or \(3\); and

• \(2\leq e_{\mathfrak {p}}\leq 2+6v_{\mathfrak {p}}(2)+3v_{\mathfrak {p}}(3)\), where \(v_{\mathfrak {p}}\) is the valuation at \(\mathfrak {p}\), if \(E\) has additive reduction and \(\mathfrak {p}\) lies above \(2\) or \(3\).

For \(\mathfrak {p}=2\) and \(3\), there is an algorithm of Tate that simultaneously creates a minimal Weierstrass equation and computes the exponent of the conductor. See:

<UL> <LI> J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33-52. <EM>Lecture Notes in Math.</EM>, Vol. <B>476</B>, Springer, Berlin, 1975.

<LI> J.H. Silverman, <EM>Advanced topics in the arithmetic of elliptic curves</EM>, GTM <B>151</B>, Springer-Verlag, New York, 1994.

</UL>

The conductor norm is the norm \([\mathcal{O}_K:\mathfrak {n}]\) of the ideal \(\mathfrak {n}\).

The discriminant \(\Delta \) of a Weierstrass equation over a field \(K\) is an element of \(K\) defined in terms of the Weierstrass coefficients. If the Weierstrass equation is

then \(\Delta \) is given by a polynomial expression in \(a_1,\dots ,a_6\), namely,

where

Then \(\Delta \neq 0\) if and only if the equation defines a smooth curve, in which case its projective closure gives an elliptic curve.

An endomorphism of an elliptic curve defined over a field \(K\) is a homomorphism \(\varphi \colon E\to E\) defined over \(K\). The set of all endomorphisms of \(E\) forms a ring called the endomorphism ring of \(E\), denoted \(\operatorname{End}(E)\), a special case of the endomorphism ring of an abelian variety.

The endomorphism ring \(\operatorname{End}(E)\) of an elliptic curve \(E\) over a field \(K\) is the ring of all endomorphisms of \(E\) defined over \(K\). For endomorphisms defined over extensions, we speak of the geometric endomorphism ring of \(E\).

For elliptic curves defined over fields of characteristic zero, this ring is isomorphic to \(\mathbb {Z}\), unless the curve has complex multiplication (CM) defined over the ground field, in which case the endomorphism ring is an order in an imaginary quadratic field; for curves defined over \(\mathbb {Q}\), this order is one of the 13 orders of class number one.

\(\operatorname{End}(E)\) always contains a subring isomorphic to \(\mathbb {Z}\), since for \(m\in \mathbb {Z}\) there is the multiplication-by-\(m\) map \([m] \colon E\to E\).

This is a special case of the endomorphism ring of an abelian variety.

If \(E\) is an elliptic curve defined over a field \(K\) and \(m\) is a positive integer, then the mod-\(m\) Galois representation attached to \(E\) is the continuous homomorphism

describing the action of the absolute Galois group of \(K\) on the \(m\)-torsion subgroup \(E[m]\).

When the characteristic of \(K\) does not divide \(m {\gt} 1\), we may identify the finite abelian group \(E[m]\) with \((\mathbb {Z}/m\mathbb {Z})^2\) and hence view the representation as a map

defined up to conjugation. In particular, when \(m=\ell \) is a prime different from the characteristic of \(K\), we have the mod-\(\ell \) Galois representation

Taking the inverse limit over prime powers \(m=\ell ^n\) yields the \(\ell \)-adic Galois representation attached to \(E\),

which describes the action of the absolute Galois group of \(K\) on \(T_{\ell }(E)\), the \(\ell \)-adic Tate module of \(E\).

When \(K\) has characteristic zero one can take the inverse limit over all positive integers \(m\) (ordered by divisibility) to obtain the adelic Galois representation

If \(E\) is an elliptic curve without complex multiplication that is defined over a number field, then the image of \(\rho _E\) is an open subgroup of \(\textrm{GL}(2,\hat{\mathbb {Z}})\) that has an associated level, index, and genus.

The image of the adelic Galois representation associate to an elliptic curve \(E\) over a number field \(K\) that does not have potential complex multiplication is an open subgroup \(H\) of \(\textrm{GL}(2,\widehat{\mathbb {Z}})\). The subgroup \(H\) has the following invariants:

• The level of \(H\) is the least positive integer \(N\) such that \(H\) is the full inverse image of its projection to \(\textrm{GL}(2,\mathbb {Z}/N\mathbb {Z})\).

• The index of \(H\) is the positive integer \([\textrm{GL}(2,\mathbb {Z}/N\mathbb {Z}):H]\).

• The genus of \(H\) is the genus of the corresponding modular curve \(X_H\).

Let \(\ell \) be a prime and let \(E\) be an elliptic curve defined over a number field \(K\).

Subgroups \(G\) of \(\textrm{GL}(2,\mathbb {F}_{\ell })\) that can arise as the image of the mod-\(\ell \) Galois representation

attached to \(E\) that do not contain \(\textrm{SL}(2,\mathbb {F}_{\ell })\) are identified using the labels introduced by Sutherland in [ . For groups with surjective determinant map (necessarily the case when \(K=\mathbb {Q}\)), these labels have the form

where \(\texttt{L}\) is the prime \(\ell \), \(\texttt{S}\) is one of G, B, Cs, Cn, Ns, Nn, A4, S4, A5, and \(\texttt{a}\), \(\texttt{b}\), \(\texttt{c}\) are optional positive integers. When the determinant map is not surjective the label has "\(\texttt{[d]}\)", where \(d\) is the index of the determinant image in \(\mathbb {F}_{\ell }^\times \).

When \(\bar\rho _{E,\ell }\) does not contain \(\textrm{SL}(2,\mathbb {F}_{\ell })\) the possibilities for \(\texttt{S}\) are: Borel B, split Cartan Cs, normalizer of the split Cartan Ns, nonsplit Cartan Cn, normalizer of the nonsplit Cartan Nn, exceptional A4, S4, A5. The cases A4 and A5 cannot occur when \(K=\mathbb {Q}\).

The geometric endomorphism ring of an elliptic curve \(E\) over a field \(K\) is \(\operatorname{End}(E_{\overline{K}})\), the endomorphism ring of the base change of \(E\) to an algebraic closure \(\overline{K}\) of \(K\).

This is a special case of the geometric endomorphism ring of an abelian variety.

A global minimal model for an elliptic curve \(E\) defined over a number field \(K\) is a Weierstrass equation for \(E\) which is integral and is a local minimal model at all primes of \(K\).

When \(K\) has class number \(1\) all elliptic curves over \(K\) have global minimal models. In general, there is an obstruction to the existence of a global minimal model for each elliptic curve \(E\) defined over \(K\), which is an ideal class of \(K\). In case this class is nontrivial for \(E\), there is a semi-global minimal model for \(E\), which is minimal at all primes except one, the ideal class of that one prime being the obstruction class.

An elliptic curve \(E\) defined over a number field \(K\) is said to have ordinary reduction at a prime \(\mathfrak {p}\) of \(K\) if the reduction \(E_{\mathfrak {p}}\) of \(E\) modulo \(\mathfrak {p}\) is smooth, and \(E_{\mathfrak {p}}\) is ordinary.

An elliptic curve \(E_{\mathfrak {p}}\) defined over a finite field of characteristic \(p\) is ordinary if \(E_{\mathfrak {p}}(\overline{\mathbb {F}_p})\) has nontrivial \(p\)-torsion.

An elliptic curve \(E\) defined over a number field \(K\) is said to have good reduction at a prime \(\mathfrak {p}\) of \(K\) if the reduction of \(E\) modulo \(\mathfrak {p}\) is smooth.

If \(E\) has good reduction at every prime of \(K\) then \(E\) is said to have everywhere good reduction.

An elliptic curve \(E\) defined over a number field \(K\) is said to have supersingular reduction at a prime \(\mathfrak {p}\) of \(K\) if the reduction \(E_{\mathfrak {p}}\) of \(E\) modulo \(\mathfrak {p}\) is smooth, and \(E_{\mathfrak {p}}\) is supersingular.

An elliptic curve \(E_{\mathfrak {p}}\) defined over a finite field of characteristic \(p\) is supersingular if \(E_{\mathfrak {p}}(\overline{\mathbb {F}_p})\) has no \(p\)-torsion.

An integral model for an elliptic curve \(E\) defined over a number field \(K\) is a Weierstrass equation for \(E\) all of whose coefficients are in the ring of integers of \(K\).

The invariants of an elliptic curve \(E\) over a number field \(K\) are its

• conductor, \(\mathfrak {N}\), which is an integral ideal of \(K\) whose norm is the conductor norm \(N(\mathfrak {N})\)

• minimal discriminant, \(\mathfrak {D}\), also an integral ideal of \(K\), whose norm is the minimal discriminant norm \(N(\mathfrak {D})\)

• j-invariant, \(j\)

• endomorphism ring, \(\text{End}(E)\)

• Sato-Tate group, \(\text{ST}(E)\)

Each Weierstrass model for \(E\) also has a discriminant, \(\Delta \), and discriminant norm, \(N(\Delta )\), which are not strictly invariants of \(E\) since different models have, in general, different discriminants.

Let \(E_1\) and \(E_2\) be two elliptic curves defined over a field \(K\). An isogeny (over \(K\)) between \(E_1\) and \(E_2\) is a non-constant morphism \(f\colon E_1 \to E_2\) defined over \(K\), i.e., a morphism of curves given by rational functions with coefficients in \(K\), such that \(f(O_{E_1})= O_{E_2}\). Elliptic curves \(E_1\) and \(E_2\) are called isogenous if there exists an isogeny \(f\colon E_1 \to E_2\).

An isogeny respects the group laws on \(E_1\) and \(E_2\), and hence determines a group homomorphism \(E_1(L)\to E_2(L)\) for any extension \(L\) of \(K\). The kernel is a finite group, defined over \(K\); in general the points in the kernel are not individually defined over \(K\) but over a finite Galois extension of \(K\) and are permuted by the Galois action.

The degree of an isogeny is its degree as a morphism of algebraic curves. For a separable isogeny this is equal to the cardinality of the kernel. Over a field of characteristic \(0\) such as a number field, all isogenies are separable. In finite characteristic \(p\), isogenies of degree coprime to \(p\) are all separable.

An isogeny is cyclic if its kernel is a cyclic group. Every isogeny is the composition of a cyclic isogeny with the multiplication-by-\(m\) map for some \(m\ge 1\).

Isogeny is an equivalence relation, and the equivalence classes are called isogeny classes. Over a number field, it is a consequence of a theorem of Shafarevich that isogeny classes are finite. Between any two curves in an isogeny class there is a unique degree of cyclic isogeny between them, except when the curves have additional endomorphisms defined over the base field of the curves; in that case there are cyclic isogenies of infinitely many different degrees between any two isogenous curves.

Isogenies from an elliptic curve \(E\) to itself are called endomorphisms. The set of all endomrpshisms of \(E\) forms a ring under pointwise addition and composition, the endomorphism ring of \(E\).

An isogeny of elliptic curves is a special case of an isogeny of abelian varieties.

The isogeny class (over a field \(K\)) of an elliptic curve \(E\) defined over \(K\) is the set of all isomorphism classes of elliptic curves defined over \(K\) that are isogenous to \(E\) over \(K\). Over a number field \(K\) this is always a finite set; over \(\mathbb {Q}\), it has at most 8 elements by a theorem of Kenku [ .

The isogeny class degree of an isogeny class of elliptic curves is the least common multiple of the degrees of all rational cyclic isogenies between elliptic curves in the isogeny class.

The isogeny graph of an isogeny class of elliptic curves is the graph whose vertices are the isomorphism classes (over the base field) of elliptic curves in the isogeny class and whose edges are the isogenies of prime degree between the curves representing the vertices.

The vertices of the isogeny graphs in the LMFDB are labeled by the final entry of the LMFDB label of the corresponding (isomorphism classes of) elliptic curves. Their edges, of which there may be several between any two given vertices, are labeled by the prime that is the degree of the corresponding isogeny.

The isogeny matrix of an isogeny class of elliptic curves is a symmetric matrix with integral entries that records the minimum among the degrees of the cyclic isogenies between the elliptic curves in the isogeny class.

In the LMFDB, the rows and columns of the matrices are ordered by the final entry of the label of the elliptic curves in the isogeny class in question, so that the \((i,j)\)-th entry is the smallest possible degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class.

An isomorphism between two elliptic curves \(E\), \(E'\) defined over a field \(K\) is an isogeny \(f:E\to E'\) such that there exist an isogeny \(g:E'\to E\) with the compositions \(g\circ f\) and \(f\circ g\) being the identity maps. Equivalently, an isomorphism \(E\to E'\) is an isogeny of degree \(1\).

Isomorphism is an equivalence relation, the equivalnce classes being called isomorphism classes.

When \(E\) and \(E'\) are defined by Weierstrass models, such an isomorphism is uniquely represented as a Weierstrass isomorphism between these models.

The \(j\)-invariant of an elliptic curve \(E\) defined over a field \(K\) is an invariant of the isomorphism class of \(E\) over \(\overline{K}\). If the Weierstrass equation of \(E\) is

then its \(j\)-invariant is given by

where

and

is the discriminant of \(E.\)

The Kodaira symbol of an elliptic curve \(E\) defined over a number field encodes the reduction type of \(E\) at a prime \(\mathfrak {p}\) of \(K\). It describes the combinatorics of the special fiber of the Né;ron model of the elliptic curve. The Né;ron model is obtained from the local minimal model for \(E\) at \(\mathfrak {p}\) using Tate’s algorithm. For an exact definition and properties, consult a text on elliptic curves.

The local data of an elliptic curve \(E\) defined over a number field \(K\) at a prime \(\mathfrak {p}\) of \(K\) consists of

• the Tamagawa number \(c_{\mathfrak {p}}\)

• the Kodaira symbol

• the reduction type

• the local root number

• the conductor valuation \(\text{ord}_{\mathfrak {p}}(\mathfrak {N})\)

• the discriminant_valuation \(\text{ord}_{\mathfrak {p}}(\mathfrak {D})\)

• the j-invariant denominator valuation \(\text{ord}_{\mathfrak {p}}(j)_{-}\)

Let \(E\) be an elliptic curve defined over a number field \(K\), and \(\mathfrak {p}\) a prime of \(K\). The local minimal discriminant of \(E\) is the ideal \(\mathfrak {p}^e\) where \(e\) is is the valuation of the discriminant of a local minimal model for \(E\) at \(\mathfrak {p}\).

A local minimal model for an elliptic curve \(E\) defined over a number field \(K\) at a prime \(\mathfrak {p}\) of \(K\) is a Weierstrass equation for \(E\) all of whose coefficients are integral at \(\mathfrak {p}\), and whose discriminant has minimal valuation at \(\mathfrak {p}\) among all such equations.

Let \(E\) be an elliptic curve over a number field \(K\), let \(\ell \) be prime, and let

be the \(\ell \)-adic Galois representation associated to \(E\).

If \(E\) does not have potential complex multiplication, then \(\rho _{E,\ell }\) is maximal if its image contains \(\textrm{SL}_2(\mathbb {Z}_{\ell })\).

In general, let \(\mathcal{O}\) be the geometric endomorphism ring of \(E\). Then \(E[\ell ^\infty ]\) is an \(\mathcal{O}\)-module, and we view \(\operatorname{Aut}_{\mathcal{O}}(E[\ell ^\infty ])\) as a subgroup of \(\operatorname{Aut}(E[\ell ^\infty ]) \simeq \textrm{GL}_2(\mathbb {Z}_{\ell })\) that contains the image of \(\rho _{E,\ell }\) whenever \(K\) contains \(\mathcal O\). We say that \(\rho _{E,\ell }\) is maximal if its image contains \(\textrm{SL}_2(\mathbb {Z}_{\ell }) \cap \operatorname{Aut}_{\mathcal{O}}(E[\ell ^\infty ])\), in which case we call \(\ell \) a maximal prime for \(E\).

Let \(E\) be an elliptic curve over a number field \(K\), let \(\ell \) be prime, and let

be the mod-\(\ell \) Galois representation associated to \(E\).

If E does not have potential complex multiplication, then \(\bar\rho _{E,\ell }\) is maximal if its image contains \(\textrm{SL}_2(\mathbb {F}_{\ell })\).

In general, let \(\mathcal{O}\) be the geometric endomorphism ring. Then \(E[\ell ]\) is an \(\mathcal{O}\)-module and we view \(\operatorname{Aut}_{\mathcal{O}}(E[\ell ]) \leq \operatorname{Aut}(E[\ell ]) \simeq \textrm{GL}_2(\mathbb {F}_{\ell })\). We say that \(\bar\rho _{E,\ell }\) is maximal if its image contains \(\textrm{SL}_2(\mathbb {F}_{\ell }) \cap \operatorname{Aut}_{\mathcal{O}}(E[\ell ])\).

For \(K=\mathbb {Q}\), the image of a maximal \(\bar\rho _{E,\ell }\) is \(\textrm{GL}_2(\mathbb {F}_{\ell })\), a Borel subgroup, the normalizer of a split Cartan subgroup, or the normalizer of a non-split Cartan subgroup, depending on whether \(\mathcal{O}=\mathbb {Z}\) or \(\mathcal{O}\ne \mathbb {Z}\) and \(\ell \) is ramified, split, or inert in \(\mathcal{O}\), respectively.

The minimal discriminant (or minimal discriminant ideal) of an elliptic curve \(E\) over a number field \(K\) is the ideal \(\mathfrak {D}_{min}\) of the ring of integers of \(K\) given by

where the product is over all primes \(\mathfrak {p}\) of \(K\), and \(\mathfrak {p}^{e_{\mathfrak {p}}}\) is the local minimal discriminant of \(E\) at \(\mathfrak {p}\).

If \(E\) has a Weierstrass model which is a global minimal model then \(\mathfrak {D}_{\mathrm{min}} = (\Delta )\), the principal ideal generated by the discriminant \(\Delta \) of this model. In general, \(\mathfrak {D}_{\mathrm{min}}\) differs from the ideal generated by the discriminant of any Weierstrass model by the 12th power of an ideal.

For an elliptic curve \(E\) defined over a field \(K\), the Mordell-Weil group of \(E/K\) is the group \(E(K)\) of \(K\)-rational points of \(E\). It is a finitely-generated Abelian group.

This is a special case of the Mordell-Weil group of an abelian variety.

The Mordell-Weil Theorem, first proved by Mordell for elliptic curves defined over \(\mathbb {Q}\) and later generalized by Weil to abelian varieties \(A\) over general number fields \(K\), states that, if \(K\) is a number field, then \(A(K)\) is a finitely generated abelian group. Its rank is called the Mordell-Weil rank of \(A\) over \(K\).

The Mordell-Weil theorem implies in particular that the torsion subgroup \(E(K)_{\mathrm{tor}}\) of \(E(K)\) is finite, and thus that the torsion order of \(E\), one of the BSD invariants, is finite.

For an elliptic curve \(E\) defined over a number field \(F\), the Mordell-Weil theorem states that the set \(E(F)\) of \(F\)-rational points on \(E\) is a finitely generated Abelian group.

This group is called the Mordell-Weil group of \(E/K\).

An elliptic curve \(E\) defined over a number field \(K\) is said to have multiplicative reduction at a prime \(\mathfrak p\) of \(K\) if the reduction of \(E\) modulo \(\mathfrak p\) has a nodal singularity.

The case of multiplicative reduction is further subdivided into split multiplicative reduction and nonsplit multiplicative reduction.

The Mordell-Weil group \(E(K)\) of an elliptic curve \(E\) over a number field \(K\) is a finitely generated abelian group, explicitly described by giving a \(\mathbb {Z}\)-basis for the group, equivalently, a (minimal) set of Mordell-Weil generators, each of which is a rational point on the curve.

The generators consist of \(r\) non-torsion generators, where \(r\) is the rank of \(E(K)\), and up to two torsion generators, which generate the torsion subgroup \(E(K)_{\textrm{tor}}\).

An elliptic curve \(E\) defined over a number field \(K\) is said to have non-split multiplicative reduction at a prime \(\mathfrak {p}\) of \(K\) if the reduction of \(E\) modulo \(\mathfrak {p}\) has a nodal singularity with tangent slopes not defined over the residue field at \(\mathfrak {p}\).

Let \(E\) be an elliptic curve defined over a number field \(K\). The obstruction class of \(E\) is an ideal class of \(K\) which is trivial if and only if \(E\) has a global minimal model.

Let \(p \in \mathbb {Z}_{\geq 0}\) be a prime and \(E\) an elliptic curve defined over a field \(K\). The \(p\)-adic Tate module of \(E\) is the inverse limit

Here for \(m\in \mathbb {Z}_{\geq 0}\), \(E[m]\) denotes the \(m\)-torsion subgroup of \(E\), which is the kernel of the multiplication-by-\(m\) endomorphism of \(E\).

If \(K\) has characteristic not equal to \(p\), then \(T_p(E)\) is a free \(\mathbb {Z}_p\)-module of rank \(2\). It carries an action of the absolute Galois group of \(K\), and thus has an associated Galois representation.

This is a special case of the Tate module of an abelian variety.

The global period \(\Omega (E/K)\) of an elliptic curve defined over a number field \(K\) is a product of local factors \(\Omega _v(E_v/K_v)\), one for each infinite place \(v\) of \(K\). Here, \(K_v\) denotes the completion of \(K\) at \(v\) (so \(K_v=\mathbb {R}\) for a real place and \(K_v=\mathbb {C}\) for a complex place), and \(E_v\) denotes the base change of \(E\) to \(K_v\).

Fixing a Weierstrass model for \(E\) with coefficients \(a_i\in K\), a model for \(E_v\) is given by the Weierstrass equation with coefficients \(a_{i,v}\), the images of \(a_i\) under \(v\) in \(K_v\). Associated to this model we have a discriminant \(\Delta (E_v)\) and an invariant differential \(\omega _v=dx/(2y+a_{1,v}x+a_{3,v})\).

For a real place given by an embedding \(v:K\to \mathbb {R}\), we define

In terms of a basis of the period lattice of \(E_v\) of the form \([x,yi]\) (when \(\Delta (E_v){\gt}0\)) or \([2x,x+yi]\) (when \(\Delta (E_v){\lt}0\)), where \(x\) and \(y\) are positive real numbers, we have \(\Omega _v(E_v)=2x\).

For a complex place given by an embedding \(v:K\to \mathbb {C}\), we define

In terms of a basis \([w_1,w_2]\) of the period lattice of \(E_v\), where \(\Im (w_2/w_1){\gt}0\), we have \(\Omega _v(E_v)=2\Im (\overline{w_1}w_2)\), which is double the covolume of the period lattice.

When \(E\) has a global minimal model, we have

In general, given an arbitrary model for \(E\) with discriminant \(\Delta (E)\), we have

where \(\mathfrak {d}\) is the minimal discriminant ideal of \(E\) and \(N(\mathfrak {d})\) denotes its norm. This quantity is independent of the model of \(E\).

An elliptic curve \(E\) defined over a number field \(K\) is said to have potential good reduction if \(E\) has everywhere good reduction over a finite extension of \(K\).

This is equivalent to the \(j\)-invariant of \(E\) being integral.

An elliptic curve \(E\) over \(\mathbb {Q}\) has a Weierstrass equation of the form

with \(a, b \in \mathbb {Z}\) such that its discriminant

Note that such an equation is not unique and \(E\) has a unique minimal Weierstrass equation.

Given a triple \(a,b,c\) of nonzero coprime integers, the quality of the triple is defined as

where \(\operatorname {rad}(abc)\) is the product of the primes dividing \(abc\). The \(abc\) conjecture stipulates that for any \(\epsilon {\gt} 0\) there are only finitely many relatively prime triples \(a,b,c\) with quality larger than \(1+\epsilon \).

The \(abc\) quality of an elliptic curve \(E\) is the quality of an \(a,b,c\) triple determined by its \(j\)-invariant, namely the one defined by writing \(\frac{j}{1728} = \frac{a}{c}\) in lowest terms and setting \(b = c - a\). Note that the \(abc\) quality is undefined for \(j=0\) and \(j=1728\).

The reason for defining the quality of \(E\) in this way comes from the equivalence of the \(abc\) conjecture with the modified Szpiro conjecture. For elliptic curves with small conductor, \(j\)-invariants often have unusually large quality.

The analytic rank of an elliptic curve \(E\) is the analytic rank of its L-function \(L(E,s)\). The weak form of the BSD conjecture implies that the analytic rank is equal to the rank of the Mordell-Weil group of \(E\).

For elliptic curves \(E\) over \(\mathbb {Q}\), it is known that \(L(E,s)\) satisfies the Hasse-Weil conjecture, and hence that the parity of the analytic rank is always compatible with the sign of the functional equation.

In general, analytic ranks stored in the LMFDB are only upper bounds on the true analytic rank (they could be incorrect if \(L(E,s)\) had a zero very close to but not on the central point). For elliptic curves over \(\mathbb {Q}\) of analytic rank less than 2 this upper bound is necessarily tight, due to parity; for analytic ranks \(2\) and \(3\) is also tight due to results of Kolyvagin; Murty and Murty; Bump, Friedberg and Hoffstein; Coates and Wiles; Gross and Zagier which together say that when the analytic rank is \(0\) or \(1\) then it equals the Mordell-Weil rank.

The Tate-Shafarevic group \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \) of an elliptic curve \(E\) defined over \(\mathbb {Q}\) is a torsion group defined in terms of Galois cohomology, which is conjectured to be finite. Its order \(\# \mathop{\mathrm{\unicode {x0428}}}\nolimits \) appears in the strong form of the Birch-Swinnerton-Dyer Conjecture for \(E\). The value of the order which is predicted by the conjecture is called the Analytic Order of Sha, \(\mathop{\mathrm{\unicode {x0428}}}\nolimits _{an}\). <!–Note that the value of \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \)_an predicted by the conjecture is always a square.–>

For elliptic curves of rank \(0\) or \(1\) it is a theorem that \(\mathop{\mathrm{\unicode {x0428}}}\nolimits \)_an is a positive rational number, and this rational number can be computed exactly; this exact computation has only been carried out for the curves in the database with rank \(0\) and conductor \(N\le 500000\). These values are always in fact integer squares in all cases computed to date. For curves of rank \(2\) and above, there is no such theorem, and the values computed are simply floating point approximate values which happen to be very close to integers. In the LMFDB we store and display the rounded values in this case.

The Birch and Swinnerton-Dyer conjecture is one of the Millennium Prize Problems listed by the Clay Mathematics Institute. It relates the order of vanishing and the first non-zero Taylor series coefficient of the L-function associated to an elliptic curve \(E\) defined over \(\mathbb {Q}\) at the central point \(s=1\) to certain arithmetic data, the BSD invariants of \(E\).

Specifically, the BSD conjecture states that the order \(r\) of vanishing of \(L(E,s)\) at \(s=1\) is equal to the rank of the Mordell-Weil group \(E(\mathbb {Q})\), and that

<!– comment: if you make the following display into a normal one using

or

then something about the html code for Sha stops it displaying properly–>

<p align="center"> \(\displaystyle \frac{1}{r!} L^{(r)}(E,1)= \displaystyle \frac{\# \mathop{\mathrm{\unicode {x0428}}}\nolimits (E/\mathbb {Q})\cdot \Omega _E \cdot \mathrm{Reg}(E/\mathbb {Q}) \cdot \prod _p c_p}{\# E(\mathbb {Q})_{\rm tor}^2}. \) </p> The quantities appearing in this formula are the BSD invariants of \(E\):

• \(r\) is the rank of \(E(\mathbb {Q})\) (a non-negative integer);

• \(\# \mathop{\mathrm{\unicode {x0428}}}\nolimits (E/\mathbb {Q})\) is the order of the Tate-Shafarevich group

of \(E\) (which is conjectured to always be finite, a positive integer);

• \(\mathrm{Reg}(E/\mathbb {Q})\) is the regulator of \(E/\mathbb {Q}\);

• \(\Omega _E\) is the real period of \(E/\mathbb {Q}\) (a positive real number);

• \(c_p\) is the Tamagawa number of \(E\) at each prime \(p\) (a positive integer which is \(1\) for all but at most finitely many primes);

• \(E(\mathbb {Q})_{\rm tor}\) is the torsion order of \(E(\mathbb {Q})\) (a positive integer).

There is a similar conjecture for abelian varieties, in which the real period is replaced by the covolume of the period lattice.

Let \(E\) be an elliptic curve defined over \(\mathbb {Q}\). The canonical height of a rational point \(P\in E(\mathbb {Q})\) is computed by writing the \(x\)-coordinate \(x(nP)=A_n(P)/D_n(P)\) as a fraction in lowest terms and setting

(<EM>Note</EM>. Some sources define \(\hat{h}\) to be \(\frac12\) of this quantity.)

Properties of \(\hat{h}\): <UL> <LI> \(\hat{h}(P)=\log \max \bigl\{ |A_1(P)|,|D_1(P)|\bigr\} +O(1)\) as \(P\) ranges over \(E(\mathbb {Q})\). <LI> \(\hat{h}(P)\ge 0\); and \(\hat{h}(P)=0\) if and only if \(P\) is a torsion point. <LI> \(\hat{h}:E(\mathbb {Q})\to \mathbb {R}\) extends to a positive definite quadratic form on \(E(\mathbb {Q})\otimes \mathbb {R}\). </UL> The height pairing on \(E\) is the associated bilinear form \(\langle P,Q\rangle =\frac{1}{2}\bigl(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)\bigr)\), which is used to compute the elliptic regulator of \(E\). It is a symmetric positive definite bilinear form on \(E(\mathbb {Q})\otimes \mathbb {R}\).

For a number field \(K\), the canonical height of \(P\in E(K)\) is given by \(\hat{h}(P)=\lim _{n\to \infty } n^{-2}h\bigl(x(nP)\bigr)\), where \(h\) is the Weil height.

The conductor \(N\) of an elliptic curve \(E\) defined over \(\mathbb {Q}\) is a positive integer divisible by the primes of bad reduction and no others. It has the form \(N=\prod p^{e_p}\), where the exponent \(e_p\) is

• \(e_p=1\) if \(E\) has multiplicative reduction at \(p\),

• \(e_p=2\) if \(E\) has additive reduction at \(p\) and \(p\ge 5\),

• \(2\leq e_p\leq 5\) if \(E\) has additive reduction and \(p=3\), and

• \(2\leq e_p\leq 8\) if \(E\) has additive reduction and \(p=2\).

For all primes \(p\), there is an algorithm of Tate that simultaneously creates a local minimal Weierstrass equation and computes the exponent of the conductor. See:

<UL> <LI> J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33-52. <EM>Lecture Notes in Math.</EM>, Vol. <B>476</B>, Springer, Berlin, 1975. [

<LI> J.H. Silverman, <EM>Advanced topics in the arithmetic of elliptic curves</EM>, GTM <B>151</B>, Springer-Verlag, New York, 1994. [

</UL>

The Cremona label of an elliptic curve over \(\mathbb {Q}\) is a way of indexing the elliptic curves over \(\mathbb {Q}.\) It has the form \(11a1\) or \(10050bf2.\) The first number represents the conductor, the letter or letters represent the isogeny class and the last number represents the isomorphism class within the isogeny class as it appears in [Cremona’s tables.](http://johncremona.github.io/ecdata/) In each isogeny class the curve with number 1 is the \(\Gamma _0(N)\)-optimal curve.<br> For more details, see "The elliptic curve database for conductors to 130000" by John Cremona in ANTS-VII proceedings, Lecture Notes in Computer Science, vol. 4076, 2006, 11-29.

In the Cremona labeling, it is somewhat difficult to describe the mechanisms for ordering isogeny classes or curves within a class, since these depend on the order in which the curves were computed (though for conductors over 230,000 the isogeny class labels coincide). Cremona labels are only available for conductors up to 500,000. For these reasons, within this site we also use the LMFDB label, whose definition is somewhat simpler. Note that the lack of internal punctuation distinguishes Cremona labels from LMFDB labels.

The discriminant \(\Delta \) of an elliptic curve \(E\) defined over \(\mathbb {Q}\) is a nonzero integer divisible exactly by the primes of bad reduction. It is the discriminant of the minimal Weierstrass equation of the curve.

The endomorphism ring \(\operatorname{End}(E)\) of an elliptic curve \(E\) is the ring of all endomorphisms of \(E\) defined over \(K\). For endomorphisms defined over extensions, we speak of the geometric endomorphism ring of \(E\).

For elliptic curves defined over \(\mathbb {Q}\), this ring is always isomorphic to \(\mathbb {Z}\) consisting of the multiplication-by-\(m\) maps \([m] \colon E\to E\) for \(m \in \mathbb {Z}\).

This is a special case of the endomorphism ring of an abelian variety.

The Faltings height of an elliptic curve \(E\) defined over \(\mathbb {Q}\) is the quantity

where \(A\) is the covolume (that is, the area of a fundamental period parallelogram) of the KNOWL(’ec.q.period_lattice’, ’Né;ron lattice’) of \(E\).

The stable Faltings height of \(E\) is

where \(j\) is the \(j\)-invariant of \(E\), \(\Delta \) the discriminant of any model of \(E\) and \(A\) the covolume of the period lattice of that model. The stable height is independent of the model of \(E\), and the unstable and stable heights are equal for semistable curves, for which \(\mathrm{denom}(j)=|\Delta |\).

In each isogeny class of elliptic curves defined over \(\mathbb {Q}\), there is a unique curve \(E_{\text{min}}\) whose KNOWL(’ec.q.period_lattice’, ’Né;ron lattice’) is a sublattice of the Né;ron lattices of all the curves in the class (G. Stevens, [ ); it is the unique curve of minimal Faltings height among the curves in the isogeny class.

The Faltings ratio of each curve \(E\) is the index of the Né;ron lattice of \(E_{\text{min}}\) in that of \(E\).

Given a triple of integers \(A, B, C\) with \(A + B = C\), the [Frey curve](https://en.wikipedia.org/wiki/Frey_curve) (or Frey-Hellegouarch curve) associated to this triple is the elliptic curve

The integral points on a given model of an elliptic curve \(E\) defined over \(\mathbb {Q}\) are the points \(P=(x,y)\) on the model that have integral coordinates \(x\) and \(y\).

The number of integral points is finite, by a theorem of Siegel.

The \(j\)-invariant of an elliptic curve \(E\) defined over \(\mathbb {Q}\) is an invariant of the isomorphism class of \(E\) over \(\overline{\mathbb {Q}}\) . If the Weierstrass equation of \(E\) is

then its \(j\)-invariant is given by

where

and

is the discriminant of \(E.\)

The Kodaira symbol encodes the reduction type of an elliptic curve at a prime \(p.\) It describes the combinatorics of the special fiber of the Né;ron model of the elliptic curve. The Né;ron model is obtained from the minimal Weierstrass equation using Tate’s algorithm. For an exact definition and properties, consult a text on elliptic curves.

The LMFDB label of an elliptic curve \(E\) over \(\mathbb {Q}\) is a way of indexing the elliptic curves over \(\mathbb {Q}.\) It has the form "11.a1" or "10050.bf2".

The label has three components: the conductor, the isogeny class label, and the isomorphism class index.

1. The first component is the decimal representation of the conductor (a positive integer).

2. The second component is the isogeny class label, a string which represents the isogeny class index, a non-negative integer encoded as in base 26 using the 26 symbols a,b,.., z. The isogeny classes of elliptic curves with the same conductor are sorted lexicographically by the \(q\)-expansions of the associated modular forms, and the isogeny class index of each isogeny class of fixed conductor is the index (starting at 0) of the class in this ordering.

3. The third component is the decimal representation of the isomorphism class index, a positive integer giving the index of the coefficient vector \([a_1, a_2, a_3, a_4, a_6]\) of the reduced minimal Weierstrass equation of \(E\) in a lexicographically sorted list of all the elliptic curves in the isogeny class.

The complete label is obtained by concatenating [conductor, ".", isogeny class label, isomorphism class index].

Note that this is not the same as the Cremona label, even though for certain curves they only differ in the insertion of the dot "." (for example, "37a1" and "37.a1" are the same curve). The presence of the punctuation "." distinguishes an LMFDB label from a Cremona label. Cremona labels are only defined for curves of conductor up to 500000.

Let \(E\) be an optimal elliptic curves of conductor \(N\), let \(f\) be the modular form associated to \(E\), and let \(\varphi :X_0(N)\to E\) be the associated modular parametrization. Let \(\omega _E\) be the Né;ron differential on \(E\). Then the pull-back \(\varphi ^*\omega _E\) of \(\omega _E\) to \(X_0(N)\) satisfies

for some non-zero rational number \(c\) called the Manin constant of \(E\). In fact \(c\in \mathbb {Z}\), by a theorem of Edixhoven.

It is conjectured that \(c=1\) for all optimal curves, and there are several results stating that \(c=1\) if certain conditions hold: see Amod Agashe, Ken Ribet and William Stein: The Manin Constant, Pure and Applied Mathematics Quarterly, Vol. 2 no.2 (2006), pp. 617–636. In an appendix to that paper, John Cremona gives an algorithm for verifying that \(c=1\) in individual cases, and proves that \(c=1\) for all optimal elliptic curves over \(\mathbb {Q}\) in the database. Kęstutis Česnavičius proves \(c=1\) for semistable elliptic curves over \(\mathbb {Q}\), and more generally that \(v_p(c) = 0\) if \(p^2 \nmid N\) in *The Manin constant in the semistable case*, Compositio Math. 154 (2018), 1889–1920.

For non-optimal elliptic curves \(E'\) over \(\mathbb {Q}\), the Manin constant is defined, in terms of the Manin constant of the unique optimal curve isogenous to \(E'\). Let \(\varphi :X_0(N)\to E\) and \(f\) be as above, and \(\psi :E\to E'\) an isogeny of least degree from \(E\) to \(E'\). Then we obtain a parametrization \(\psi \circ \varphi :X_0(N)\to E'\) and define the Manin constant \(c'\) of \(E'\) to be the non-zero rational number such that

This is an integer multiple of the Manin constant of \(E\), since \(\psi ^*\omega _{E'}\) is an integer multiple of \(\omega _E\); the multiplier divides the degree of \(\psi \) but may be strictly less: it may equal \(1\).

The minimal quadratic twist of an elliptic curve \(E\) defined over \(\mathbb {Q}\) is defined as follows.

• First consider the finite set of all quadratic twists of \(E\) which have minimal conductor. If this set contains just one curve, it is the minimal quadratic twist.

• Otherwise, sort the curves with minimal conductor into isogeny classes, and restrict attention to the curves whose class comes first in the LMFDB labelling; equivalently, sort the curves by the sequence of coefficients \((a_n)\) of their \(L\)-function and restrict to the curve or curves with the first such sequence.

• If \(E\) does not have Complex Multiplication (CM), then the minimal isogeny class contains a *unique* curve with the same \(j\)-invariant as \(E\), and this curve is the minimal quadratic twist of \(E\).

• If \(E\) does have CM, then the minimal isogeny class contains exactly *two* curves with \(j\)-invariant \(j(E)\). In all but one case these two curves have distinct minimal discriminants, with the same sign, and we define the minimal quadratic twist to be the curve whose minimal discriminant has smallest absolute value.

• The exception is for elliptic curves with \(j=66^3\), which have CM by the imaginary quadratic order with discriminant \(-16\). The minimal conductor is \(32\), and curves [32.a1](https://www.lmfdb.org/EllipticCurve/Q/32/a/1) and [32.a2](https://www.lmfdb.org/EllipticCurve/Q/32/a/2) (which are quadratic twists of each other by \(-1\)) both have minimal discriminant \(2^9\). The minimal quadratic twist for \(j=66^3\) is defined to be [32.a1](https://www.lmfdb.org/EllipticCurve/Q/32/a/1).

All elliptic curves \(E\) over \(\mathbb {Q}\) with \(j\)-invariant \(1728\) are quartic twists of each other. The smallest conductor of such a curve is \(32\). Both the curves [32.a3](https://www.lmfdb.org/EllipticCurve/Q/32/a/3) and [32.a4](https://www.lmfdb.org/EllipticCurve/Q/32/a/4) have \(j\)-invariant \(1728\), and they have minimal discriminants \(-2^{12}\) and \(2^6\) respectively. We define the minimal quartic twist (or just minimal twist) of every elliptic curve with \(j=1728\) to be the curve [32.a3](https://www.lmfdb.org/EllipticCurve/Q/32/a/3), which has smaller discriminant, and equation \(Y^2=X^3-X\).

All elliptic curves \(E\) over \(\mathbb {Q}\) with \(j\)-invariant \(0\) are sextic twists of each other. The smallest conductor of such a curve is \(27\). Both the curves [27.a3](https://www.lmfdb.org/EllipticCurve/Q/27/a/3) and [27.a4](https://www.lmfdb.org/EllipticCurve/Q/27/a/4) have \(j\)-invariant \(0\), and they have minimal discriminants \(-3^9\) and \(-3^3\) respectively. We define the minimal sextic twist (or just minimal twist) of every elliptic curve with \(j=0\) to be the curve [27.a4](https://www.lmfdb.org/EllipticCurve/Q/27/a/4), which has smaller discriminant, and equation \(Y^2+Y=X^3\).

The minimal twist of an elliptic curve \(E\) is its minimal quadratic twist, unless \(j(E)=0\) or \(1728\), in which cases the minimal twist is its minimal sextic or quartic twist respectively. The minimal quadratic twist depends only on the \(j\)-invariant unless \(j=0\) or \(1728\); in each of these cases, there are infinitely many different minimal quadratic twists, though only one minimal twist.

Every elliptic curve over \(\mathbb {Q}\) has an integral Weierstrass model (or equation) of the form

where \(a_1,a_2,a_3,a_4,a_6\) are integers. Each such equation has a discriminant \(\Delta \). A minimal Weierstrass equation is one for which \(|\Delta |\) is minimal among all Weierstrass models for the same curve. For elliptic curves over \(\mathbb {Q}\), minimal models exist, and there is a unique reduced minimal model which satisfies the additional constraints \(a_1,a_3\in \{ 0,1\} \), \(a_2\in \{ -1,0,1\} \).

The modular degree of an elliptic curve over \(\mathbb {Q}\) is the minimum degree of a modular parametrization of the curve.

A modular parametrization of an elliptic curve \(E\) over \(\mathbb {Q}\) is a non-constant map \(X_0(N) \to E,\) where \(N\) is the conductor of \(E.\)

The naive height of an elliptic curve in short Weierstrass form

is the quantity \(\max (4\lvert a_4 \rvert ^3, 27\lvert a_6 \rvert ^2)\).

An elliptic curve over \(\mathbb {Q}\) is optimal if it is an optimal quotient of the corresponding modular curve. Every isogeny class contains a unique optimal curve. For more information, see [William Stein’s page on optimal quotients.](http://wstein.org/papers/ars-manin/html/node2.html)

Optimal curves have a Cremona label whose last component is the number 1, with the exception of class 990h where the optimal curve is 990h3 (number 3). This is a historical accident and has no mathematical significance.

NB It has not yet been proved in all cases that the first curve in each class is optimal; however this is true for all isogeny classes of conductor \({}\le 400000\), and for many others (for example whenever the isogeny class consists of only one curve). The current optimality status of each curve is shown on its home page.

For \(E\) an elliptic curve defined over \(\mathbb {C}\) by a Weierstrass equation with coefficients \(a_1,a_2,a_3,a_4,a_6\), the period lattice of \(E\) is the set \(\Lambda \) of periods of the invariant differential \(dx/(2y+a_1x+a_3)\), which is a discrete lattice of rank \(2\) in \(\mathbb {C}\). There is an isomorphism (of complex Lie groups) \(\mathbb {C}/\Lambda \cong E(\mathbb {C})\) defined in terms of the Weierstrass \(\wp \)-function.

For elliptic curves defined over \(\mathbb {R}\) (and in particular, for those defined over \(\mathbb {Q}\)), the period lattice has one of two possible types dependng on the sign of the discriminant \(\Delta \) of \(E\):

• If \(\Delta {\gt}0\), then \(\Lambda \) is *rectangular*, with a \(\mathbb {Z}\)-basis of the form \(\left\langle x,yi\right\rangle \), where \(x\) and \(y\) are positive real numbers; in this case, \(E(\mathbb {R})\) has two connected components.

• If \(\Delta {\lt}0\), then \(\Lambda \) has a \(\mathbb {Z}\)-basis of the form \(\left\langle 2x,x+yi\right\rangle \), where \(x\) and \(y\) are positive real numbers; in this case, \(E(\mathbb {R})\) has one connected component.

The real period of \(E\) is defined to be \(2x\) in each case, so is equal to the smallest positive real period multiplied by the number of real components.

Note that the period lattice depends on the choice of Weierstrass model of \(E\); different models have homothetic lattices. For elliptic curves defined over \(\mathbb {Q}\), the period lattice associated to a global minimal model of \(E\) is called the Né;ron lattice of \(E\). The real period of the Né;ron lattice is denoted \(\Omega _E\), and appears in the Birch Swinnerton-Dyer conjecture for \(E\).

For an elliptic curve \(E\) defined over \(\mathbb {R}\) with period lattice \(\Lambda \), the real period \(\Omega \) is the least positive element of \(\Lambda \cap \mathbb {R}\) multiplied by the number of components of \(E(\mathbb {R})\).

When an elliptic curve is defined by means of a Weierstrass equation, the period lattice \(\Lambda \) is the lattice of periods of the invariant differential \(dx/(2y+a_1x+a_3)\). Different Weierstrass models defining isomorphic curves have period lattices which are homothetic, meaning that they differ by a nonzero multiplicative constant. When we speak of the period lattice or the real period for an elliptic curve defined over \(\mathbb {Q}\), we always mean the lattice and period associated with a minimal equation.

The reduction type of an elliptic curve \(E\) defined over \(\mathbb {Q}\) at a prime \(p\) depends on the reduction \(\tilde E\) of \(E\) modulo \(p\). This reduction is constructed by taking a minimal Weierstrass equation for \(E\) and reducing its coefficients modulo \(p\) to obtain a curves over \(\mathbb {F}_p\). The reduced curve is either smooth (non-singular) or has a unique singular point.

\(E\) has good reduction at \(p\) if \(\tilde E\) is non-singular over \(\mathbb {F}_p\). The reduction type is ordinary (ord) if \(\tilde E\) is ordinary (equivalently, if \(\tilde E(\overline{\mathbb {F}_p})\) has non-trivial \(p\)-torsion) and supersingular (ss) otherwise. The coefficient \(a(p)\) of the L-function \(L(E,s)\) is divisible by \(p\) if the reduction is supersingular and not if it is ordinary.

\(E\) has bad reduction at \(p\) if \(\tilde E\) is singular over \(\mathbb {F}_p\). In this case the reduction type is further classified according to the nature of the singularity. In all cases the singularity is a double point.

\(E\) has multiplicative reduction at \(p\) if \(\tilde E\) has a nodal singularity: the singular point is a node, with distinct tangents. It is called split if the two tangents are defined over \(\mathbb {F}_p\) and non-split otherwise. The coefficient \(a(p)\) of \(L(E,s)\) is \(1\) if the reduction is split and \(-1\) if it is non-split.

\(E\) has additive reduction at \(p\) if \(\tilde E\) has a cuspidal singularity: the singular point is a cusp, with only one tangent. In this case \(a(p)=0\).

The regulator of an elliptic curve \(E\) defined over a number field \(K,\) denoted \(\operatorname {Reg}(E/K)\), is the volume of \(E(K)/E(K)_{tor}\) with respect to the height pairing \(\langle -,-\rangle \) associated to the canonical height \(\hat{h}\), i.e. \(\langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))\).

If the Mordell-Weil group \(E(K)\) has rank \(r\) and \(P_1, \ldots , P_r \in E(K)\) generate \(E(K)/E(K)_{tor}\), then

which is independent of the choice of generators.

Special cases are when \(E(K)\) has rank \(0\), in which case \(E(K)/E(K)_{tor}=0\) and \(\operatorname {Reg}(E/K)=1\), and when \(E(K)\) has rank \(1\), in which case \(\operatorname {Reg}(E/K)\) is equal to the canonical height \(\hat{h}(P)\) of a generator \(P\).

An elliptic curve is semistable if it has multiplicative reduction at every bad prime.

Let \(\bar\rho _{E,\ell }\) be the mod-\(\ell \) Galois representation of an elliptic curve \(E/\mathbb {Q}\).

The Serre invariants \((k,M)\) of \(\bar\rho _{E,\ell }\) consist of the Serre weight \(k\) and the Serre conductor \(M\) giving the weight and minimal level of a newform \(f\in S_{k}^{\textrm new}(\Gamma _1(M))\) whose associated mod-\(\ell \) Galois representation is isomorphic to \(\bar\rho _{E,\ell }\).

This means that \(a_p(E)\) and \(a_p(f)\) reduce to the same element of the residue field of a prime above \(\ell \) in the coefficient field of \(f\) (this residue field need not have degree one, but every \(a_p(f)\) must reduce to an element of \(\mathbb {F}_{\ell }\) in order for this condition to hold).

The modular form \(f\) is not uniquely determined, but the minimal level \(M\) arising among all such \(f\) is uniquely determined, and among those with level \(M\), the weight is uniquely determined.

For all but finitely many primes \(\ell \), including all \(\ell {\gt}7\) of good reduction for \(E\), the Serre invariants are \((2,N)\), where \(N\) is the conductor of the elliptic curve. The primes \(\ell \) for which this does not hold are exceptional.

In general, the Serre weight \(k\) is divisible by \(2\) and the Serre conductor \(M\) divides \(N\).

The special value of an elliptic curve \(E/\mathbb {Q}\) is the first nonzero value of \(L^{(r)}(E,1)/r!\) for \(r\in \mathbb {Z}_{\ge 0}\), where \(L(E,s)\) is the \(L\)-function of \(E\) in its arithmetic normalization.

The special value appears on the LHS of the formula in the Birch and Swinnerton-Dyer conjecture.

The (modified) Szpiro ratio of an elliptic curve \(E\) is defined as

where \(N\) is the conductor of \(E\) and \(c_4\) and \(c_6\) are defined as for the \(j\)-invariant. The (modified) Szpiro conjecture is that, for any \(\epsilon {\gt} 0\), there are only finitely many elliptic curves with Szpiro ratio larger than \(6+\epsilon \). In [ , Oesterlé proves that this conjecture is equivalent to the \(abc\) conjecture.

In Oesterlé’s paper cited above, there is another conjecture, that the ratio

also has the property of only taking values larger than \(6+\epsilon \) finitely many times (here \(\Delta \) is the minimal discriminant of \(E\)). This conjecture is implied by the modified Szpiro conjecture (and thus the \(abc\) conjecture), but it is not currently known to be equivalent. All of the Szpiro ratios in the LMFDB are computed in terms of \(c_4\) and \(c_6\) rather than \(\Delta \) for this reason.

Let \(E\) be an elliptic curve defined over \(\mathbb {Q}\) and let \(K\) be a number field. We say that there is torsion growth from \(\mathbb {Q}\) to \(K\) if the torsion subgroup \(E(K)_{\rm tor}\) of \(E(K)\) is strictly larger than \(E(\mathbb {Q})_{\rm tor}\).

If there is torsion growth in a field \(K\) then obviously the torsion also grows in every extension of \(K\). We say that the torsion growth in \(K\) is primitive if \(E(K)_{\rm tor}\) is strictly larger than \(E(K')_{\rm tor}\) for all proper subfields \(K' \subsetneq K\).

For every elliptic curve \(E\) there is torsion growth in at least one field of degree \(2\), \(3\), or \(4\), and torsion can only grow in fields whose degree is divisible by \(2\), \(3\), \(5\) or \(7\): see Theorem 7.2 of [ . Additionally, there is no primitive torsion growth in fields of degrees \(22\) or \(26\): see Lemma 2.11 of [ . Hence the only degrees less than \(24\) in which primitive torsion growth occurs are \(2,3,4,5,6,7,8,9,10,12,14,15,16,18,20,21\).

If \(E\) is an elliptic curve defined over \(\mathbb {Q}\), its torsion subgroup is the subgroup of the Mordell-Weil group \(E(\mathbb {Q})\) consisting of all the rational points of finite order. It is a finite abelian group of order at most \(16\) (by a theorem of Mazur), which is a product of at most \(2\) cyclic factors. The "torsion structure" is the list of invariants of the group:

• \([]\) for the trivial group;

• \([n]\) for a cyclic group of order \(n\) (only \(n=2,3,4,5,6,7,8,9,10\) or \(12\) occur for elliptic curves over \(\mathbb {Q}\));

• \([n_1,n_2]\) with \(n_1\mid n_2\) for a product of cyclic groups of orders \(n_1\) and \(n_2\) (only \([2,2m]\) for \(m=2,4,6\) or \(8\) occur over \(\mathbb {Q}\)).

An elliptic curve \(E\) defined over a number field \(K\) is a \(\mathbb {Q}\)-curve if it is isogenous over \(\overline{K}\) to each of its Galois conjugates. Note that the isogenies need not be defined over \(K\) itself.

An elliptic curve which is the base change of a curve defined over \(\mathbb {Q}\) is a \(\mathbb {Q}\)-curve, but not all \(\mathbb {Q}\)-curves are base-change curves.

Elliptic curves with CM are all \(\mathbb {Q}\)-curves, as are all those whose \(j\)-invariant is in \(\mathbb {Q}\).

The rank of an elliptic curve \(E\) defined over a number field \(K\) is the rank of its Mordell-Weil group \(E(K)\).

The Mordell-Weil Theorem says that \(E(K)\) is a finitely-generated abelian group, hence

where \(E(K)_{\rm tor}\) is the finite torsion subgroup of \(E(K)\), and \(r\geq 0\) is the rank.

Rank is an isogeny invariant: all curves in an isogeny class have the same rank.

An elliptic curve \(E\) over a number field \(K\) is semistable if it has multiplicative reduction at every bad prime, and has potential good reduction if its \(j\)-invariant is integral.

If \(E\) has potential good reduction then it cannot be semistable unless it has everywhere good reduction.

The reduction type of an elliptic curve \(E\) defined over a number field \(K\) at a prime \(\mathfrak {p}\) of \(K\) depends on the reduction \(\tilde E\) of \(E\) modulo \(\mathfrak {p}\). Let \(\mathbb {F}_q\) be the ring of integers of \(K\) modulo \(\mathfrak {p}\), a finite field of characteristic \(p\).

\(E\) has good reduction at \(\mathfrak p\) if \(\tilde E\) is non-singular over \(\mathbb {F}_q\). The reduction type is ordinary if \(\tilde E\) is ordinary (equivalently, \(\tilde E(\overline{\mathbb {F}_q})\) has \(p\)-torsion) and supersingular otherwise.

On the other hand, if the reduction of \(E\) modulo \(\mathfrak {p}\) is singular, then \(E\) has bad reduction. There are two types of bad reduction are as follows.

\(E\) has multiplicative reduction at \(\mathfrak p\) if \(\tilde E\) has a nodal singularity. It is called split multiplicative reduction if the two tangents at the node are defined over \(\mathbb {F}_q\) and non-split multiplicative reduction otherwise.

\(E\) has additive reduction at \(\mathfrak p\) if \(\tilde E\) has a cuspidal singularity.

The regulator of an elliptic curve \(E\) defined over a number field \(K\), denoted \(\operatorname {Reg}(E/K)\), is the volume of \(E(K)/E(K)_{tor}\) with respect to the height pairing \(\langle -,-\rangle \) associated to the canonical height \(\hat{h}\), i.e. \(\langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))\).

If the Mordell-Weil group \(E(K)\) has rank \(r\) and \(P_1, \ldots , P_r \in E(K)\) generate \(E(K)/E(K)_{\mathrm{tor}}\), then

which is independent of the choice of generators.

Special cases are when \(E(K)\) has rank \(0\), in which case \(E(K)/E(K)_{\mathrm{tor}}=0\) and \(\operatorname {Reg}(E/K)=1\), and when \(E(K)\) has rank \(1\), in which case \(\operatorname {Reg}(E/K)\) is equal to the canonical height \(\hat{h}(P)\) of a generator \(P\).

The canonical height used to define the regulator is usually *normalised* so that it is invariant under base change. Note that the regulator which appears in the Birch Swinnerton-Dyer conjecture is with respect to the non-normalised height; this is sometimes called the Néron-Tate regulator, and denoted \(\operatorname {Reg}_{\rm NT}(E/K)\). These are related by

where \(d\) is the degree \([K:\mathbb {Q}]\).

An elliptic curve over a commutative ring \(R\) is an elliptic scheme \(E \to \operatorname {Spec} R\).

For example, an elliptic curve over \(\mathbb {Z}[1/N]\) is the same as an elliptic curve over \(\mathbb {Q}\) with good reduction at all primes not dividing \(N\). (More precisely, the latter is the generic fiber of the former.)

An elliptic scheme over a scheme \(S\) is a smooth proper morphism \(E \to S\) whose fibers are elliptic curves.

An elliptic curve \(E\) defined over a number field \(K\) of class number \(h(K)\) greater than \(1\) may not have a global minimal model. In this case there still exist semi-global minimal models for \(E\) which are local minimal models at all except one prime. At this prime, the discriminant valuation exceeds that of the minimal discriminant ideal by \(12\).

An elliptic curve is semistable if it has multiplicative reduction at every bad prime.

Every elliptic curve over a field \(k\) whose characteristic is not 2 or 3 has a simplified equation (or short Weierstrass model) of the form \(y^2 = x^3 + Ax + B\). When \(k=\mathbb {Q}\) is the field of rational numbers, one can choose \(A\) and \(B\) to be integers.

For elliptic curves over \(\mathbb {Q}\) this model will necessarily have bad reduction at 2, even when \(E\) has good reduction at 2; it may also bad reduction at 3 even when the minimal model of \(E\) does not.

The special value of an elliptic curve \(E\) defined over a number field \(K\) is the first nonzero value of \(L^{(r)}(E,1)/r!\) for \(r\in \mathbb {Z}_{\ge 0}\), where \(L(E/K,s)\) is the L-function of \(E\) in its arithmetic normalization. It is also known as the leading coefficient of the L-function.

The special value appears in the Birch and Swinnerton-Dyer conjecture.

An elliptic curve \(E\) defined over a number field \(K\) is said to have split multiplicative reduction at a prime \(\mathfrak {p}\) of \(K\) if the reduction of \(E\) modulo \(\mathfrak {p}\) has a nodal singularity with both tangent slopes defined over the residue field at \(\mathfrak {p}\).

The Tamagawa number of an elliptic curve \(E\) defined over a number field at a prime \(\mathfrak {p}\) of \(K\) is the index \([E(K_{\mathfrak {p}}):E^0(K_{\mathfrak {p}})]\), where \(K_{\mathfrak {p}}\) is the completion of \(K\) at \(\mathfrak {p}\) and \(E^0(K_{\mathfrak {p}})\) is the subgroup of \(E(K_{\mathfrak {p}})\) consisting of all points whose reduction modulo \(\mathfrak {p}\) is smooth.

The Tamagawa number of \(E\) at \(\mathfrak {p}\) is usually denoted \(c_{\mathfrak {p}}(E)\). It is a positive integer, and equal to \(1\) if \(E\) has good reduction at \(\mathfrak {p}\) and may be computed in general using Tate’s algorithm.

The product of the Tamagawa numbers over all primes is a positive integer known as the Tamagawa product.

The torsion order of an elliptic curve \(E\) over a field \(K\) is the order of the torsion subgroup \(E(K)_{\text{tor}}\) of its Mordell-Weil group E(K).

The torsion subgroup \(E(K)_{\text{tor}}\) is the set of all points on \(E\) with coordinates in \(K\) having finite order in the group \(E(K)\). When \(K\) is a number field (for example, when \(K=\mathbb {Q}\)) it is a finite set, since by the Mordell-Weil Theorem, \(E(K)\) is finitely generated.

When \(K=\mathbb {Q}\) the torsion order \(n\) satisfies \(n\le 16\), by a theorem of Mazur.

For an elliptic curve \(E\) over a field \(K,\) the torsion subgroup of \(E\) over \(K\) is the subgroup \(E(K)_{\text{tor}}\) of the Mordell-Weil group \(E(K)\) consisting of points of finite order. For a number field \(K\) this is always a finite group, since by the Mordell-Weil Theorem \(E(K)\) is finitely generated.

The torsion subgroup is always either cyclic or a product of two cyclic groups. The torsion structure is the list of invariants of the group:

• \([]\) for the trivial group;

• \([n]\) for a cyclic group of order \(n{\gt}1\);

• \([n_1,n_2]\) with \(n_1\mid n_2\) for a product of non-trivial cyclic groups of orders \(n_1\) and \(n_2\).

For \(K=\mathbb {Q}\) the possible torsion structures are \([n]\) for \(n\le 10\) and \(n=12\), and \([2,2n]\) for \(n=1,2,3,4\).

A twist of an elliptic curve \(E\) defined over a field \(K\) is another elliptic curve \(E'\), also defined over \(K\), which is isomorphic to \(E\) over the algebraic closure of \(K\).

Two elliptic curves are twists if an only if they have the same \(j\)-invariant.

For elliptic curves \(E\) with \(j(E)\not=0, 1728\), the only twists of \(E\) are its quadratic twists \(E^{(d)}\). Provided that the characteristic of \(K\) is not \(2\), the nontrivial quadratic twists of \(E\) are in bijection with the nontrivial elements \(d\) of \(K^*/(K^*)^2\), and \(E^{(d)}\) is isomorphic to \(E\) over the quadratic extension \(K(\sqrt{d})\).

Over fields of characteristic not \(2\) or \(3\), elliptic curves with \(j\)-invariant \(1728\) also admit quartic twists, parametrised by \(K^*/(K^*)^4\), and elliptic curves with \(j\)-invariant \(0\) also admit sextic twists, parametrised by \(K^*/(K^*)^6\). Elliptic curves \(E\) over fields \(K\) of characteristic \(2\) and \(3\) with \(j(E)=0=1728\) have nonabelian automorphism groups, and their twists are more complicated to describe, being in all cases parametrised by \(H^1(\operatorname{Gal}(\overline{K}/K), \operatorname{Aut}(E))\).

Elliptic curve twists are a special case of twists of abelian varieties.

A Weierstrass equation or Weierstrass model over a field \(k\) is a plane curve \(E\) of the form

with \(a_1, a_2, a_3, a_4, a_6 \in k\).

The Weierstrass coefficients of this model \(E\) are the five coefficients \(a_i\). These are often displayed as a list \([a_1, a_2, a_3, a_4, a_6]\).

It is common not to distinguish between the affine curve defined by a Weierstrass equation and its projective closure, which contains exactly one additional point at infinity, \([0:1:0]\).

A Weierstrass model is smooth if and only if its discriminant \(\Delta \) is nonzero. In this case, the plane curve \(E\) together with the point at infinity as base point, define an elliptic curve defined over \(k\).

Two smooth Weierstrass models define isomorphic elliptic curves if and only if they are isomorphic as Weierstrass models.

Two Weierstrass models \(E\), \(E'\) over a field \(K\) with Weierstrass coefficients \([a_1,a_2,a_3,a_4,a_6]\) and \([a'_1,a'_2,a'_3,a'_4,a'_6]\) are isomorphic over \(K\) if there exist \(u\in K^*\) and \(r,s,t\in K\) such that

The set of transformations with parameters \([u,r,s,t]\in K^*\times K^3\) form the group of Weierstrass isomorphisms, which acts on both the set of all Weierstrass models over \(K\) and also on the subset of smooth models, preserving the point at infinity. The discriminants \(\Delta \), \(\Delta '\) of the two models are related by

In the smooth case such a Weierstrass isomorphism \([u,r,s,t]\) induces an isomorphism between the two elliptic curves \(E\), \(E'\) they define. In terms of affine coordinates this is given by

where