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([-5,-16,14,23,0,-3]),K([-7,-21,14,29,1,-4]),K([3,3,-5,-7,0,1]),K([-21,116,3,-160,-17,23]),K([94,-227,-63,268,37,-39])])
gp: E = ellinit([Polrev([-5,-16,14,23,0,-3]),Polrev([-7,-21,14,29,1,-4]),Polrev([3,3,-5,-7,0,1]),Polrev([-21,116,3,-160,-17,23]),Polrev([94,-227,-63,268,37,-39])], K);
magma: E := EllipticCurve([K![-5,-16,14,23,0,-3],K![-7,-21,14,29,1,-4],K![3,3,-5,-7,0,1],K![-21,116,3,-160,-17,23],K![94,-227,-63,268,37,-39]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((6a^5-2a^4-43a^3-17a^2+29a+5)\) | = | \((6a^5-2a^4-43a^3-17a^2+29a+5)\) |
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: | \((18a^5-6a^4-128a^3-51a^2+79a+21)\) | = | \((6a^5-2a^4-43a^3-17a^2+29a+5)^{2}\) |
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: | \( \frac{2634370822463}{5041} a^{5} - \frac{740439838413}{5041} a^{4} - \frac{19177442690303}{5041} a^{3} - \frac{8090290411651}{5041} a^{2} + \frac{13243147162542}{5041} a + \frac{3945985807101}{5041} \) | ||
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 | $\left(3 a^{5} - a^{4} - 21 a^{3} - 10 a^{2} + 12 a + 8 : -3 a^{5} + 5 a^{4} + 16 a^{3} - 14 a^{2} - a : 1\right)$ |
| 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: | $\left(-\frac{3}{4} a^{5} + \frac{3}{2} a^{4} + 4 a^{3} - \frac{25}{4} a^{2} - \frac{5}{4} a + \frac{9}{2} : \frac{15}{4} a^{5} - \frac{19}{8} a^{4} - 25 a^{3} - \frac{9}{2} a^{2} + 14 a + \frac{11}{4} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((6a^5-2a^4-43a^3-17a^2+29a+5)\) | \(71\) | \(2\) | \(I_{2}\) | 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
71.4-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.