Base field \(\Q(\zeta_{21})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -8, 8, 6, -6, -1, 1]))
gp: K = nfinit(Polrev([1, -8, 8, 6, -6, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 8, 6, -6, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,6,-2,-5,1,1]),K([2,-9,-4,6,1,-1]),K([0,0,0,0,0,0]),K([-2634,3192,1978,-2027,-358,298]),K([-162582,208990,125585,-143434,-21647,22934])])
gp: E = ellinit([Polrev([-1,6,-2,-5,1,1]),Polrev([2,-9,-4,6,1,-1]),Polrev([0,0,0,0,0,0]),Polrev([-2634,3192,1978,-2027,-358,298]),Polrev([-162582,208990,125585,-143434,-21647,22934])], K);
magma: E := EllipticCurve([K![-1,6,-2,-5,1,1],K![2,-9,-4,6,1,-1],K![0,0,0,0,0,0],K![-2634,3192,1978,-2027,-358,298],K![-162582,208990,125585,-143434,-21647,22934]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-3a-1)\) | = | \((a^3+a^2-3a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 43 \) | = | \(43\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((379a^5-689a^4-2667a^3+3058a^2+4051a-2005)\) | = | \((a^3+a^2-3a-1)^{11}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 929293739471222707 \) | = | \(43^{11}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{122671686822496847013039511249758605577398}{929293739471222707} a^{5} + \frac{80040521603412429280297685957868481938607}{929293739471222707} a^{4} - \frac{603764956031188829103770286089077775728189}{929293739471222707} a^{3} - \frac{261677914170315830376360951082281856697755}{929293739471222707} a^{2} + \frac{548956616586298989964223373216163970685955}{929293739471222707} a - \frac{74235009600386253636048203807121847689780}{929293739471222707} \) | ||
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: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 0.045066886011824715087865461697045371927 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.979493 \) | ||
| Analytic order of Ш: | \( 14641 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-3a-1)\) | \(43\) | \(1\) | \(I_{11}\) | Non-split multiplicative | \(1\) | \(1\) | \(11\) | \(11\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(11\) | 11B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
11.
Its isogeny class
43.1-b
consists of curves linked by isogenies of
degree 11.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.