Base field \(\Q(\sqrt{2}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))
gp: K = nfinit(Polrev([-2, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([-7344,-4590]),K([-321908,-220212])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([-7344,-4590]),Polrev([-321908,-220212])], K);
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![-7344,-4590],K![-321908,-220212]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((36)\) | = | \((a)^{4}\cdot(3)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 1296 \) | = | \(2^{4}\cdot9^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((69984a)\) | = | \((a)^{11}\cdot(3)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -9795520512 \) | = | \(-2^{11}\cdot9^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{123062343233293457}{3} a + 58012144939294440 \) | ||
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 | $\left(\frac{91656}{529} a + \frac{600395}{2116} : \frac{398288675}{97336} a + \frac{553327731}{97336} : 1\right)$ |
| Height | \(8.6341909174174329154952269325943769424\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-36 a - \frac{35}{2} : \frac{35}{4} a + 36 : 1\right)$ |
| 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: | \( 8.6341909174174329154952269325943769424 \) | ||
| Period: | \( 0.19377332241295022998050243543503452433 \) | ||
| Tamagawa product: | \( 8 \) = \(2\cdot2^{2}\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.3660865726789108095675585240999577417 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(I_{3}^{*}\) | Additive | \(-1\) | \(4\) | \(11\) | \(0\) |
| \((3)\) | \(9\) | \(4\) | \(I_{1}^{*}\) | Additive | \(1\) | \(2\) | \(7\) | \(1\) |
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
1296.1-b
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.