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([1,0]),K([-1,0]),K([0,1]),K([-54,-1004]),K([11592,-11736])])
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,1]),Polrev([-54,-1004]),Polrev([11592,-11736])], K);
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![-54,-1004],K![11592,-11736]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((153a)\) | = | \((a)\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(-2a+3)\cdot(2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 46818 \) | = | \(2\cdot3^{2}\cdot3^{2}\cdot17\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-8873883720a+45290937294)\) | = | \((a)^{2}\cdot(-a-1)^{18}\cdot(a-1)^{10}\cdot(-2a+3)^{2}\cdot(2a+3)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2208760625521202119236 \) | = | \(2^{2}\cdot3^{18}\cdot3^{10}\cdot17^{2}\cdot17^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{208995993887450}{44386483761} a + \frac{92858826184591}{88772967522} \) | ||
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 | $\left(8 a - 11 : -2 a + 182 : 1\right)$ | |
| Height | \(1.5821252141944329466086040028408485846\) | |
| 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: | $\left(-15 a - 16 : 7 a + 8 : 1\right)$ | $\left(-6 a - \frac{61}{4} : \frac{5}{2} a + \frac{61}{8} : 1\right)$ |
| 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: | \( 1.5821252141944329466086040028408485846 \) | ||
| Period: | \( 0.21222479839262587408841647059963404518 \) | ||
| Tamagawa product: | \( 128 \) = \(2\cdot2^{2}\cdot2^{2}\cdot2\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.7987609628167022574030743488021917380 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((-a-1)\) | \(3\) | \(4\) | \(I_{12}^{*}\) | Additive | \(-1\) | \(2\) | \(18\) | \(12\) |
| \((a-1)\) | \(3\) | \(4\) | \(I_{4}^{*}\) | Additive | \(-1\) | \(2\) | \(10\) | \(4\) |
| \((-2a+3)\) | \(17\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((2a+3)\) | \(17\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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
46818.8-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.