Base field \(\Q(\sqrt{-31}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 8 \); class number \(3\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -1, 1]))
gp: K = nfinit(Polrev([8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([8,-2]),K([4,0])])
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([0,1]),Polrev([8,-2]),Polrev([4,0])], K);
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![8,-2],K![4,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a)\) | = | \((2,a)^{4}\cdot(2,a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 32 \) | = | \(2^{4}\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-224a-256)\) | = | \((2,a)^{14}\cdot(2,a+1)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 524288 \) | = | \(2^{14}\cdot2^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{514073}{32} a - 48120 \) | ||
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: | \(0\) |
| 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: | $\left(-\frac{1}{4} a : -\frac{3}{8} a - 1 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 3.6845144366370951848609099777180877933 \) | ||
| Tamagawa product: | \( 4 \) = \(2^{2}\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.3235166564284357065964412379544104894 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2,a)\) | \(2\) | \(4\) | \(I_{6}^{*}\) | Additive | \(-1\) | \(4\) | \(14\) | \(2\) |
| \((2,a+1)\) | \(2\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
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 |
| \(5\) | 5B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 5 and 10.
Its isogeny class
32.2-a
consists of curves linked by isogenies of
degrees dividing 10.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.