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,-2,0,1,0,0]),K([3,-2,-2,4,0,-1]),K([1,0,-3,0,1,0]),K([3,-6,4,2,-2,0]),K([1,-4,5,0,-3,1])])
gp: E = ellinit([Polrev([1,-2,0,1,0,0]),Polrev([3,-2,-2,4,0,-1]),Polrev([1,0,-3,0,1,0]),Polrev([3,-6,4,2,-2,0]),Polrev([1,-4,5,0,-3,1])], K);
magma: E := EllipticCurve([K![1,-2,0,1,0,0],K![3,-2,-2,4,0,-1],K![1,0,-3,0,1,0],K![3,-6,4,2,-2,0],K![1,-4,5,0,-3,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5-a^4+5a^3+3a^2-5a-1)\) | = | \((-a^5-a^4+5a^3+3a^2-5a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^5+6a^3-9a+2)\) | = | \((-a^5-a^4+5a^3+3a^2-5a-1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -41 \) | = | \(-41\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{13444954}{41} a^{5} - \frac{11843590}{41} a^{4} - \frac{81028162}{41} a^{3} + \frac{68125115}{41} a^{2} + \frac{114760190}{41} a - \frac{87740634}{41} \) | ||
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: | \( 603.94057741679419555345677352454947071 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.896535 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5-a^4+5a^3+3a^2-5a-1)\) | \(41\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 41.6-d consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.