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([1,1]),K([1,1]),K([0,0]),K([-1230,-465]),K([14840,5609])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,0]),Polrev([-1230,-465]),Polrev([14840,5609])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,0],K![-1230,-465],K![14840,5609]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((18a+54)\) | = | \((a+3)^{3}\cdot(-a+2)^{2}\cdot(-a-2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 648 \) | = | \(2^{3}\cdot3^{2}\cdot3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((26244)\) | = | \((a+3)^{4}\cdot(-a+2)^{8}\cdot(-a-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 688747536 \) | = | \(2^{4}\cdot3^{8}\cdot3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{35152}{9} \) | ||
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 | $\left(-7 a - 18 : -41 a - 108 : 1\right)$ | |
| Height | \(0.40943857783737939430220140934424334316\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-11 a - 29 : 20 a + 53 : 1\right)$ | $\left(\frac{5}{2} a + 7 : -\frac{19}{4} a - \frac{49}{4} : 1\right)$ |
| 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: | \( 0.40943857783737939430220140934424334316 \) | ||
| Period: | \( 12.401492634428814718752155867842209557 \) | ||
| Tamagawa product: | \( 32 \) = \(2\cdot2^{2}\cdot2^{2}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.8383422403053732066954961709488466164 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(2\) | \(2\) | \(III\) | Additive | \(-1\) | \(3\) | \(4\) | \(0\) |
| \((-a+2)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
| \((-a-2)\) | \(3\) | \(4\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
648.1-f
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
| Base field | Curve |
|---|---|
| \(\Q\) | 144.b4 |
| \(\Q\) | 3528.d4 |