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([0,0]),K([0,0]),K([-115,0]),K([392,0])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-115,0]),Polrev([392,0])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![-115,0],K![392,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((195)\) | = | \((-a-1)\cdot(a-1)\cdot(5)\cdot(13)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 38025 \) | = | \(3\cdot3\cdot25\cdot169\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((27720225)\) | = | \((-a-1)^{8}\cdot(a-1)^{8}\cdot(5)^{2}\cdot(13)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 768410874050625 \) | = | \(3^{8}\cdot3^{8}\cdot25^{2}\cdot169^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{168288035761}{27720225} \) | ||
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 | $\left(6 a + 11 : -36 a - 22 : 1\right)$ | |
| Height | \(1.3204585285748150828193093051581977348\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(4 : -2 : 1\right)$ | $\left(-1 : -22 : 1\right)$ |
| 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: | \( 1.3204585285748150828193093051581977348 \) | ||
| Period: | \( 0.74126833486817411528296423277758685098 \) | ||
| Tamagawa product: | \( 256 \) = \(2^{3}\cdot2^{3}\cdot2\cdot2\) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 5.5370086712880988228516247432958443620 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a-1)\) | \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((a-1)\) | \(3\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
| \((5)\) | \(25\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((13)\) | \(169\) | \(2\) | \(I_{2}\) | 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 |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
38025.2-a
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 |
|---|---|
| \(\Q\) | 195.a5 |
| \(\Q\) | 12480.ca5 |