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,-19,13,29,1,-4]),K([5,32,-16,-49,-3,7]),K([-5,-19,13,29,1,-4]),K([354,1180,-861,-1864,-67,259]),K([-11664,-39489,26921,60616,2341,-8418])])
gp: E = ellinit([Polrev([-5,-19,13,29,1,-4]),Polrev([5,32,-16,-49,-3,7]),Polrev([-5,-19,13,29,1,-4]),Polrev([354,1180,-861,-1864,-67,259]),Polrev([-11664,-39489,26921,60616,2341,-8418])], K);
magma: E := EllipticCurve([K![-5,-19,13,29,1,-4],K![5,32,-16,-49,-3,7],K![-5,-19,13,29,1,-4],K![354,1180,-861,-1864,-67,259],K![-11664,-39489,26921,60616,2341,-8418]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+a^3+6a^2-a-4)\) | = | \((-a^4+a^3+6a^2-a-4)\) |
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: | \((-15a^5+4a^4+108a^3+49a^2-71a-17)\) | = | \((-a^4+a^3+6a^2-a-4)^{2}\) |
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: | \( -\frac{61875465519471921}{5041} a^{5} + \frac{186718135324916475}{5041} a^{4} + \frac{56396294925773264}{5041} a^{3} - \frac{237534589193567280}{5041} a^{2} + \frac{46132023113659095}{5041} a + \frac{30668266783208944}{5041} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(138 a^{5} - 40 a^{4} - 995 a^{3} - 430 a^{2} + 663 a + 195 : -1638 a^{5} + 453 a^{4} + 11793 a^{3} + 5256 a^{2} - 7661 a - 2265 : 1\right)$ |
| 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: | \( 12723.959494764252722620040840363918388 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 1.29032 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+6a^2-a-4)\) | \(71\) | \(2\) | \(I_{2}\) | Non-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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
71.6-c
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.