Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,2,1,-4,0,1]),K([0,0,1,4,0,-1]),K([-2,1,4,-4,-1,1]),K([-1,-4,4,7,2,0]),K([-1,1,-2,-1,6,3])])
gp: E = ellinit([Polrev([-2,2,1,-4,0,1]),Polrev([0,0,1,4,0,-1]),Polrev([-2,1,4,-4,-1,1]),Polrev([-1,-4,4,7,2,0]),Polrev([-1,1,-2,-1,6,3])], K);
magma: E := EllipticCurve([K![-2,2,1,-4,0,1],K![0,0,1,4,0,-1],K![-2,1,4,-4,-1,1],K![-1,-4,4,7,2,0],K![-1,1,-2,-1,6,3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^5-a^4-10a^3+3a^2+9a-1)\) | = | \((a^4-4a^2+a+2)\cdot(-a^3+3a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 91 \) | = | \(7\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-3a^5+14a^4+10a^3-46a^2-a-9)\) | = | \((a^4-4a^2+a+2)^{7}\cdot(-a^3+3a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -139178767 \) | = | \(-7^{7}\cdot13^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{75920719558025}{139178767} a^{5} + \frac{26110568103036}{139178767} a^{4} + \frac{396790644808426}{139178767} a^{3} - \frac{43416547811453}{139178767} a^{2} - \frac{408239914338619}{139178767} a - \frac{115757315220418}{139178767} \) | ||
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 | $\left(-a^{5} + a^{4} + 6 a^{3} - 5 a^{2} - 11 a + 7 : -17 a^{5} + 27 a^{4} + 75 a^{3} - 110 a^{2} - 38 a + 48 : 1\right)$ |
| Height | \(0.39368797569078567703849921334951135545\) |
| Torsion structure: | \(\Z/14\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-9 a^{5} + 12 a^{4} + 41 a^{3} - 51 a^{2} - 30 a + 29 : -65 a^{5} + 90 a^{4} + 299 a^{3} - 370 a^{2} - 212 a + 205 : 1\right)$ |
| 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: | \( 0.39368797569078567703849921334951135545 \) | ||
| Period: | \( 15349.554745232716667031555037539968963 \) | ||
| Tamagawa product: | \( 14 \) = \(7\cdot2\) | ||
| Torsion order: | \(14\) | ||
| Leading coefficient: | \( 3.36409 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-4a^2+a+2)\) | \(7\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
| \((-a^3+3a)\) | \(13\) | \(2\) | \(I_{2}\) | 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 |
| \(7\) | 7B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
91.1-b
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.