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([1,2,0,-4,0,1]),K([-1,8,-3,-6,1,1]),K([1,0,-3,0,1,0]),K([1,-3,3,1,-1,0]),K([0,0,0,0,0,0])])
gp: E = ellinit([Polrev([1,2,0,-4,0,1]),Polrev([-1,8,-3,-6,1,1]),Polrev([1,0,-3,0,1,0]),Polrev([1,-3,3,1,-1,0]),Polrev([0,0,0,0,0,0])], K);
magma: E := EllipticCurve([K![1,2,0,-4,0,1],K![-1,8,-3,-6,1,1],K![1,0,-3,0,1,0],K![1,-3,3,1,-1,0],K![0,0,0,0,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-a^4-4a^3+4a^2+4a-2)\) | = | \((a^5-a^4-4a^3+4a^2+4a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 53 \) | = | \(53\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-5a^4-3a^3+18a^2+6a-7)\) | = | \((a^5-a^4-4a^3+4a^2+4a-2)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -148877 \) | = | \(-53^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{50593425026}{148877} a^{5} - \frac{14755246679}{148877} a^{4} - \frac{263310198039}{148877} a^{3} + \frac{15768359082}{148877} a^{2} + \frac{314421209160}{148877} a + \frac{71068595640}{148877} \) | ||
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 | $\left(3 a^{5} - 7 a^{4} - 7 a^{3} + 23 a^{2} - 7 a - 3 : -20 a^{5} + 43 a^{4} + 51 a^{3} - 139 a^{2} + 39 a + 17 : 1\right)$ |
| Height | \(0.0012791912557583409003467950630334646415\) |
| 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: | \( 0.0012791912557583409003467950630334646415 \) | ||
| Period: | \( 60162.441045792087048848477885082004309 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.27340 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-a^4-4a^3+4a^2+4a-2)\) | \(53\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
53.6-a
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.