Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,14,5,-12,-1,2]),K([-2,12,6,-12,-1,2]),K([1,7,0,-6,0,1]),K([-5,37,15,-35,-1,5]),K([-3,19,1,-17,6,0])])
gp: E = ellinit([Polrev([-1,14,5,-12,-1,2]),Polrev([-2,12,6,-12,-1,2]),Polrev([1,7,0,-6,0,1]),Polrev([-5,37,15,-35,-1,5]),Polrev([-3,19,1,-17,6,0])], K);
magma: E := EllipticCurve([K![-1,14,5,-12,-1,2],K![-2,12,6,-12,-1,2],K![1,7,0,-6,0,1],K![-5,37,15,-35,-1,5],K![-3,19,1,-17,6,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-3a^5+2a^4+17a^3-9a^2-15a)\) | = | \((-3a^5+2a^4+17a^3-9a^2-15a)\) |
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: | \((-a^5+a^4+6a^3-6a^2-7a+5)\) | = | \((-3a^5+2a^4+17a^3-9a^2-15a)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 71 \) | = | \(71\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1497412}{71} a^{5} + \frac{2670984}{71} a^{4} + \frac{7853018}{71} a^{3} - 205878 a^{2} - \frac{4974801}{71} a + \frac{11806486}{71} \) | ||
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 | $\left(7 a^{5} - 5 a^{4} - 41 a^{3} + 23 a^{2} + 43 a - 5 : -42 a^{5} + 28 a^{4} + 248 a^{3} - 132 a^{2} - 264 a + 31 : 1\right)$ |
| Height | \(0.22095699775900382026933421020447041490\) |
| Torsion structure: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{3} + a^{2} + 2 a : 2 a^{5} - 3 a^{4} - 8 a^{3} + 8 a^{2} + 8 a - 2 : 1\right)$ |
| 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: | \( 0.22095699775900382026933421020447041490 \) | ||
| Period: | \( 41839.532971909374145793439845048731629 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 2.64531 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-3a^5+2a^4+17a^3-9a^2-15a)\) | \(71\) | \(1\) | \(I_{1}\) | 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 |
|---|---|
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
71.2-d
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.