Base field 6.6.1134389.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 6 x^{3} + 4 x^{2} - 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 4, 6, -4, -2, 1]))
gp: K = nfinit(Polrev([-1, -3, 4, 6, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 4, 6, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,1,0,0,0,0]),K([3,0,-13,5,5,-2]),K([4,2,-9,2,3,-1]),K([-23,0,48,-15,-16,6]),K([12,6,-27,5,9,-3])])
gp: E = ellinit([Polrev([0,1,0,0,0,0]),Polrev([3,0,-13,5,5,-2]),Polrev([4,2,-9,2,3,-1]),Polrev([-23,0,48,-15,-16,6]),Polrev([12,6,-27,5,9,-3])], K);
magma: E := EllipticCurve([K![0,1,0,0,0,0],K![3,0,-13,5,5,-2],K![4,2,-9,2,3,-1],K![-23,0,48,-15,-16,6],K![12,6,-27,5,9,-3]]);
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: | \((-a^5+2a^4+4a^3-7a^2-3a+5)\) | = | \((a^3-a^2-2a+1)\) |
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: | \( \frac{124200}{13} a^{5} - \frac{154732}{13} a^{4} - \frac{604704}{13} a^{3} + \frac{256238}{13} a^{2} + \frac{701798}{13} a + \frac{201965}{13} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/7\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a^{5} - a^{4} - 5 a^{3} + 6 a + 5 : 3 a^{5} - 4 a^{4} - 14 a^{3} + 7 a^{2} + 15 a + 5 : 1\right)$ |
| 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: | \( 86211.576181876407179242211159434063711 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(7\) | ||
| Leading coefficient: | \( 1.65192 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-2a+1)\) | \(13\) | \(1\) | \(I_{1}\) | Non-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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
13.1-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.