Base field \(\Q(\sqrt{7}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, 0, 1]))
gp: K = nfinit(Polrev([-7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,0]),K([-306,0]),K([-5859,0])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,0]),Polrev([0,0]),Polrev([-306,0]),Polrev([-5859,0])], K);
magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,0],K![-306,0],K![-5859,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((9a)\) | = | \((-a+2)^{2}\cdot(-a-2)^{2}\cdot(a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 567 \) | = | \(3^{2}\cdot3^{2}\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-12607619787)\) | = | \((-a+2)^{7}\cdot(-a-2)^{7}\cdot(a)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 158952076693553925369 \) | = | \(3^{7}\cdot3^{7}\cdot7^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{4354703137}{17294403} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(245 a + \frac{1345}{2} : -\frac{37605}{4} a - \frac{99519}{4} : 1\right)$ |
| Height | \(2.6385088231341581748855033900597912662\) |
| Torsion structure: | \(\Z/4\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(60 : -30 a - 441 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 2.6385088231341581748855033900597912662 \) | ||
| Period: | \( 0.27134014530509803619669937469932143172 \) | ||
| Tamagawa product: | \( 256 \) = \(2^{2}\cdot2^{2}\cdot2^{4}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 4.3295580470557848261672342392333532909 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+2)\) | \(3\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
| \((-a-2)\) | \(3\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
| \((a)\) | \(7\) | \(16\) | \(I_{16}\) | Split multiplicative | \(-1\) | \(1\) | \(16\) | \(16\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 8 and 16.
Its isogeny class
567.1-c
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 441.f4 |
| \(\Q\) | 1008.l4 |