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([-2,2,1,-4,0,1]),K([4,0,-5,0,1,0]),K([2,0,-4,0,1,0]),K([-7,82,-132,24,43,-16]),K([-82,-296,894,-308,-279,127])])
gp: E = ellinit([Polrev([-2,2,1,-4,0,1]),Polrev([4,0,-5,0,1,0]),Polrev([2,0,-4,0,1,0]),Polrev([-7,82,-132,24,43,-16]),Polrev([-82,-296,894,-308,-279,127])], K);
magma: E := EllipticCurve([K![-2,2,1,-4,0,1],K![4,0,-5,0,1,0],K![2,0,-4,0,1,0],K![-7,82,-132,24,43,-16],K![-82,-296,894,-308,-279,127]]);
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: | \( -\frac{1827330655296306713158}{6241} a^{5} + \frac{2267851397132794939683}{6241} a^{4} + \frac{8589934719036282821459}{6241} a^{3} - \frac{9380127050470680153907}{6241} a^{2} - \frac{8702685181704913513870}{6241} a + \frac{7579977528109156697231}{6241} \) | ||
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 | $\left(2 a^{5} - 5 a^{4} - 5 a^{3} + 16 a^{2} - 5 a : 7 a^{5} - 17 a^{4} - 15 a^{3} + 56 a^{2} - 22 a - 7 : 1\right)$ |
| Height | \(0.038527218061788762413655668891111412804\) |
| 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: | $\left(\frac{1}{4} a^{5} - \frac{3}{2} a^{4} + \frac{3}{4} a^{3} + \frac{17}{4} a^{2} - 5 a + \frac{3}{4} : -\frac{3}{4} a^{5} + \frac{3}{2} a^{4} + 3 a^{3} - \frac{19}{4} a^{2} - \frac{13}{8} a + \frac{13}{8} : 1\right)$ |
| 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: | \(1\) | ||
| Regulator: | \( 0.038527218061788762413655668891111412804 \) | ||
| Period: | \( 13265.049995389334293006990187993138672 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.51617 \) | ||
| 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 \( p < 1000 \) 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.
Its isogeny class
79.2-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.