Base field 6.6.1279733.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -11, 10, 10, -6, -2, 1]))
gp: K = nfinit(Polrev([-1, -11, 10, 10, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 10, 10, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-2,5,-2,-2,1]),K([-2,1,-1,3,1,-1]),K([0,1,0,0,0,0]),K([-758,713,429,-384,-57,48]),K([-8105,7542,5920,-4688,-1072,724])])
gp: E = ellinit([Polrev([-1,-2,5,-2,-2,1]),Polrev([-2,1,-1,3,1,-1]),Polrev([0,1,0,0,0,0]),Polrev([-758,713,429,-384,-57,48]),Polrev([-8105,7542,5920,-4688,-1072,724])], K);
magma: E := EllipticCurve([K![-1,-2,5,-2,-2,1],K![-2,1,-1,3,1,-1],K![0,1,0,0,0,0],K![-758,713,429,-384,-57,48],K![-8105,7542,5920,-4688,-1072,724]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^4+a^3+4a^2-a-1)\) | = | \((-a^4+a^3+4a^2-a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 7 \) | = | \(7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^5+2a^4+3a^3-6a^2+3)\) | = | \((-a^4+a^3+4a^2-a-1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -49 \) | = | \(-7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1307780824025356872024765295948965037006665607}{7} a^{5} - \frac{1426930782360125700490308291091787017129909047}{7} a^{4} - \frac{9143610193366820283442749807521617239815916698}{7} a^{3} + \frac{4767258720016713504152715749572825371641060421}{7} a^{2} + \frac{17410729135899459869092701189140814182089683462}{7} a + \frac{1438874534605637964988709222541326697190256371}{7} \) | ||
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: | \(0 \le r \le 5\) |
| 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: | $\left(-\frac{13}{4} a^{5} - a^{4} + 29 a^{3} - 5 a^{2} - \frac{107}{2} a + \frac{117}{4} : -\frac{1}{8} a^{5} + \frac{57}{8} a^{4} - \frac{15}{4} a^{3} - 29 a^{2} + \frac{47}{8} a + \frac{113}{8} : 1\right)$ |
| 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: | \(0 \le r \le 5\) | ||
| Regulator: | not available | ||
| Period: | \( 0.95645328078739093283928501436968604879 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.62611 \) | ||
| Analytic order of Ш: | not available | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^4+a^3+4a^2-a-1)\) | \(7\) | \(2\) | \(I_{2}\) | 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, 4 and 8.
Its isogeny class
7.2-d
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.