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(Pol(Vecrev([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([-110976,0]),K([-14248080,0])])
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-110976,0])),Pol(Vecrev([-14248080,0]))], K);
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![-110976,0],K![-14248080,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((204)\) | = | \((a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41616 \) | = | \(2^{4}\cdot3\cdot3\cdot17\cdot17\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((332928)\) | = | \((a)^{14}\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(-2a+3)^{2}\cdot(2a+3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 110841053184 \) | = | \(2^{14}\cdot3^{2}\cdot3^{2}\cdot17^{2}\cdot17^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2361739090258884097}{5202} \) | ||
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 | $\left(-\frac{32532}{169} : \frac{211356}{2197} a : 1\right)$ |
| Height | \(1.91349088474424\) |
| 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: | $\left(-193 : 88 a : 1\right)$ |
| 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: | \( 1.91349088474424 \) | ||
| Period: | \( 0.0918770736059740 \) | ||
| Tamagawa product: | \( 64 \) = \(2^{2}\cdot2\cdot2\cdot2\cdot2\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 3.97803437985983 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(4\) | \(I_{6}^{*}\) | Additive | \(1\) | \(4\) | \(14\) | \(2\) |
| \((-a-1)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((a-1)\) | \(3\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
| \((-2a+3)\) | \(17\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((2a+3)\) | \(17\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
41616.5-g
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This curve is the base change of 3264.m1, 816.b1, defined over \(\Q\), so it is also a \(\Q\)-curve.