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,-2,1,1,0,0]),K([-1,-5,4,5,-1,-1]),K([-1,1,1,0,0,0]),K([1,-7,0,7,0,-1]),K([0,-2,3,-2,-2,2])])
gp: E = ellinit([Polrev([-1,-2,1,1,0,0]),Polrev([-1,-5,4,5,-1,-1]),Polrev([-1,1,1,0,0,0]),Polrev([1,-7,0,7,0,-1]),Polrev([0,-2,3,-2,-2,2])], K);
magma: E := EllipticCurve([K![-1,-2,1,1,0,0],K![-1,-5,4,5,-1,-1],K![-1,1,1,0,0,0],K![1,-7,0,7,0,-1],K![0,-2,3,-2,-2,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4+a^3-5a^2-3a+3)\) | = | \((a^4+a^3-5a^2-3a+3)\) |
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: | \((-a^4+a^3+4a^2-3a-1)\) | = | \((a^4+a^3-5a^2-3a+3)\) |
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: | \( -\frac{7678148}{41} a^{5} - \frac{7305801}{41} a^{4} + \frac{31443377}{41} a^{3} + \frac{15419812}{41} a^{2} - \frac{29392241}{41} a + \frac{4246630}{41} \) | ||
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4+a^3-5a^2-3a+3)\) | \(41\) | \(1\) | \(I_{1}\) | 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 41.2-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.