Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,10,5,-11,-1,2]),K([0,7,1,-6,0,1]),K([-2,0,1,0,0,0]),K([0,0,-2,-1,2,0]),K([-2,9,0,-10,3,1])])
gp: E = ellinit([Polrev([-1,10,5,-11,-1,2]),Polrev([0,7,1,-6,0,1]),Polrev([-2,0,1,0,0,0]),Polrev([0,0,-2,-1,2,0]),Polrev([-2,9,0,-10,3,1])], K);
magma: E := EllipticCurve([K![-1,10,5,-11,-1,2],K![0,7,1,-6,0,1],K![-2,0,1,0,0,0],K![0,0,-2,-1,2,0],K![-2,9,0,-10,3,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+a^4+12a^3-4a^2-13a)\) | = | \((-2a^5+a^4+12a^3-4a^2-13a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 13 \) | = | \(13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^5+5a^4+24a^3-25a^2-30a+13)\) | = | \((-2a^5+a^4+12a^3-4a^2-13a)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 371293 \) | = | \(13^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{341618999984}{371293} a^{5} - \frac{652186405930}{371293} a^{4} - \frac{1777962603488}{371293} a^{3} + \frac{3602250670634}{371293} a^{2} + \frac{1073690215744}{371293} a - \frac{3003492585005}{371293} \) | ||
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 | $\left(-3 a^{5} + 19 a^{3} - a^{2} - 25 a + 3 : -13 a^{5} + 12 a^{4} + 67 a^{3} - 54 a^{2} - 38 a + 6 : 1\right)$ |
| Height | \(0.0019232170070705003710675348394286047507\) |
| 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: | \( 0.0019232170070705003710675348394286047507 \) | ||
| Period: | \( 31237.662803809166217890140102450431887 \) | ||
| Tamagawa product: | \( 5 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.14881 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+12a^3-4a^2-13a)\) | \(13\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
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 13.3-a 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.