Base field 3.3.257.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -4, -1, 1]))
gp: K = nfinit(Polrev([3, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1]),K([-4,-1,1]),K([-3,0,1]),K([-5,9,-5]),K([-18,31,-12])])
gp: E = ellinit([Polrev([-3,0,1]),Polrev([-4,-1,1]),Polrev([-3,0,1]),Polrev([-5,9,-5]),Polrev([-18,31,-12])], K);
magma: E := EllipticCurve([K![-3,0,1],K![-4,-1,1],K![-3,0,1],K![-5,9,-5],K![-18,31,-12]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2+a-6)\) | = | \((a^2-3)\cdot(a^2-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 21 \) | = | \(3\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-40a^2-55a+426)\) | = | \((a^2-3)^{8}\cdot(a^2-2)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 15752961 \) | = | \(3^{8}\cdot7^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2469125117705}{15752961} a^{2} + \frac{2848919682052}{15752961} a - \frac{3194204179760}{15752961} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(2 a - 1 : -a^{2} - a + 3 : 1\right)$ | $\left(-2 a + 3 : -a^{2} + a + 3 : 1\right)$ |
| 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: | \( 33.450923160087644535395384858288273243 \) | ||
| Tamagawa product: | \( 8 \) = \(2\cdot2^{2}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.0433056285174515055920384841410256533 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-3)\) | \(3\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((a^2-2)\) | \(7\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
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 and 4.
Its isogeny class
21.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.