Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([1,1]),K([-1394,892]),K([2249,-11606])])
gp: E = ellinit([Polrev([0,1]),Polrev([-1,0]),Polrev([1,1]),Polrev([-1394,892]),Polrev([2249,-11606])], K);
magma: E := EllipticCurve([K![0,1],K![-1,0],K![1,1],K![-1394,892],K![2249,-11606]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((210a-144)\) | = | \((-2a+1)^{2}\cdot(2)\cdot(6a-5)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 34596 \) | = | \(3^{2}\cdot4\cdot31^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-157308971616a+58461864480)\) | = | \((-2a+1)^{6}\cdot(2)^{5}\cdot(6a-5)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 18967326369258620722176 \) | = | \(3^{6}\cdot4^{5}\cdot31^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{511363962461}{916132832} a + \frac{764718499383}{458066416} \) | ||
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: | 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: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 0.19102298247265229136718134721579665526 \) | ||
| Tamagawa product: | \( 10 \) = \(1\cdot5\cdot2\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.2057434070398192930900589403494535044 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a+1)\) | \(3\) | \(1\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
| \((2)\) | \(4\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
| \((6a-5)\) | \(31\) | \(2\) | \(I_{5}^{*}\) | Additive | \(-1\) | \(2\) | \(11\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5Cs.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
34596.3-d
consists of curves linked by isogenies of
degrees dividing 25.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.