Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-12,-39,27,58,2,-8]),K([1,18,-7,-28,-2,4]),K([-7,-26,18,37,1,-5]),K([1,-8,-24,45,11,-8]),K([-27,-97,131,96,-15,-10])])
gp: E = ellinit([Polrev([-12,-39,27,58,2,-8]),Polrev([1,18,-7,-28,-2,4]),Polrev([-7,-26,18,37,1,-5]),Polrev([1,-8,-24,45,11,-8]),Polrev([-27,-97,131,96,-15,-10])], K);
magma: E := EllipticCurve([K![-12,-39,27,58,2,-8],K![1,18,-7,-28,-2,4],K![-7,-26,18,37,1,-5],K![1,-8,-24,45,11,-8],K![-27,-97,131,96,-15,-10]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+a^4+14a^3+4a^2-9a-4)\) | = | \((-2a^5+a^4+14a^3+4a^2-9a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3a^5+21a^3+17a^2-7a-7)\) | = | \((-2a^5+a^4+14a^3+4a^2-9a-4)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -5041 \) | = | \(-71^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2420952621619}{5041} a^{5} - \frac{464004317169}{5041} a^{4} - \frac{246254617652}{71} a^{3} - \frac{9103530423609}{5041} a^{2} + \frac{11054899265248}{5041} a + \frac{4395816073739}{5041} \) | ||
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 | $\left(-13 a^{5} + 3 a^{4} + 94 a^{3} + 45 a^{2} - 59 a - 15 : 21 a^{5} - 4 a^{4} - 152 a^{3} - 79 a^{2} + 92 a + 28 : 1\right)$ |
| Height | \(0.0060775655434875208948068184142481340551\) |
| 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: | $\left(a^{5} - \frac{1}{4} a^{4} - \frac{13}{2} a^{3} - \frac{17}{4} a^{2} + a + 3 : -\frac{1}{2} a^{5} + a^{4} + \frac{19}{8} a^{3} - \frac{27}{8} a^{2} - \frac{1}{4} a + \frac{3}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.0060775655434875208948068184142481340551 \) | ||
| Period: | \( 66963.547141587985708763348767015196595 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.22863 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+14a^3+4a^2-9a-4)\) | \(71\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
71.2-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.