Base field 6.6.1683101.1
Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 4 x^{4} + 13 x^{3} + 7 x^{2} - 14 x - 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, -14, 7, 13, -4, -3, 1]))
gp: K = nfinit(Polrev([-7, -14, 7, 13, -4, -3, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, -14, 7, 13, -4, -3, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,2,5,-3,-2,1]),K([-6,-8,3,3,-1,0]),K([3,9,2,-6,-1,1]),K([39,104,11,-79,-13,15]),K([128,311,9,-223,-25,38])])
gp: E = ellinit([Polrev([-1,2,5,-3,-2,1]),Polrev([-6,-8,3,3,-1,0]),Polrev([3,9,2,-6,-1,1]),Polrev([39,104,11,-79,-13,15]),Polrev([128,311,9,-223,-25,38])], K);
magma: E := EllipticCurve([K![-1,2,5,-3,-2,1],K![-6,-8,3,3,-1,0],K![3,9,2,-6,-1,1],K![39,104,11,-79,-13,15],K![128,311,9,-223,-25,38]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-2a+1)\) | = | \((a^3-a^2-2a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 13 \) | = | \(13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^2+2a+2)\) | = | \((a^3-a^2-2a+1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -13 \) | = | \(-13\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{118442868691}{13} a^{5} - \frac{490545599991}{13} a^{4} + \frac{86259780319}{13} a^{3} + \frac{1441261639769}{13} a^{2} - \frac{816322283539}{13} a - \frac{726245075566}{13} \) | ||
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 | $\left(-5 a^{5} + 7 a^{4} + 24 a^{3} - 15 a^{2} - 28 a + 1 : 23 a^{5} - 25 a^{4} - 125 a^{3} + 39 a^{2} + 164 a + 38 : 1\right)$ |
| Height | \(0.11008540647655866931861795712909752520\) |
| 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: | \( 0.11008540647655866931861795712909752520 \) | ||
| Period: | \( 6041.7648426958759066105322027360822794 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.07602 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-2a+1)\) | \(13\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.6.1[3] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
13.2-a
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.