Base field \(\Q(\zeta_{13})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,6,-3,-5,1,1]),K([-1,1,-3,0,1,0]),K([3,-2,-4,1,1,0]),K([8,16,-27,-35,7,9]),K([-13,-37,45,71,-8,-16])])
gp: E = ellinit([Polrev([0,6,-3,-5,1,1]),Polrev([-1,1,-3,0,1,0]),Polrev([3,-2,-4,1,1,0]),Polrev([8,16,-27,-35,7,9]),Polrev([-13,-37,45,71,-8,-16])], K);
magma: E := EllipticCurve([K![0,6,-3,-5,1,1],K![-1,1,-3,0,1,0],K![3,-2,-4,1,1,0],K![8,16,-27,-35,7,9],K![-13,-37,45,71,-8,-16]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-4a^3-a^2+2a+3)\) | = | \((a^5-4a^3-a^2+2a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 79 \) | = | \(79\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^4-a^3-10a^2+2a+7)\) | = | \((a^5-4a^3-a^2+2a+3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6241 \) | = | \(79^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{58199203663}{6241} a^{5} + \frac{29929897535}{6241} a^{4} - \frac{246476187924}{6241} a^{3} - \frac{141422125490}{6241} a^{2} + \frac{137785461801}{6241} a + \frac{37929353079}{6241} \) | ||
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 | $\left(2 a^{5} - a^{4} - 12 a^{3} + a^{2} + 17 a + 6 : 9 a^{5} - 2 a^{4} - 47 a^{3} + a^{2} + 58 a + 16 : 1\right)$ |
| Height | \(0.0011713697873235358072686064267420056249\) |
| 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: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.0011713697873235358072686064267420056249 \) | ||
| Period: | \( 107610.22356721157453697223324609030866 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.48239 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-4a^3-a^2+2a+3)\) | \(79\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 79.2-b 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.