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([1,1]),K([1,1]),K([0,1]),K([-20,-2]),K([-54,8])])
gp: E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-20,-2]),Polrev([-54,8])], K);
magma: E := EllipticCurve([K![1,1],K![1,1],K![0,1],K![-20,-2],K![-54,8]]);
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: | \((991440a-396576)\) | = | \((a)^{9}\cdot(-a-1)^{6}\cdot(a-1)^{9}\cdot(-2a+3)\cdot(2a+3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2123179070976 \) | = | \(2^{9}\cdot3^{6}\cdot3^{9}\cdot17\cdot17\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{226719}{544} a + \frac{33345}{272} \) | ||
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 | $\left(-a - 5 : -3 a + 1 : 1\right)$ |
| Height | \(1.1309245602392836663407276135936290951\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 1.1309245602392836663407276135936290951 \) | ||
| Period: | \( 1.2951672232311818958204118395866324435 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot1\cdot2\cdot1\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 4.1429002276323372858660461761687848605 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(1\) | \(I_{9}\) | Non-split multiplicative | \(1\) | \(1\) | \(9\) | \(9\) |
| \((-a-1)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((a-1)\) | \(3\) | \(2\) | \(III^{*}\) | Additive | \(1\) | \(2\) | \(9\) | \(0\) |
| \((-2a+3)\) | \(17\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
| \((2a+3)\) | \(17\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 46818.8-f consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.