Base field 6.6.1279733.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -11, 10, 10, -6, -2, 1]))
gp: K = nfinit(Polrev([-1, -11, 10, 10, -6, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -11, 10, 10, -6, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,0,1,0,0,0]),K([-2,5,6,-4,-2,1]),K([-2,-3,6,-2,-2,1]),K([-744,535,706,-347,-151,65]),K([-102152,78787,91318,-50436,-18199,8693])])
gp: E = ellinit([Polrev([-2,0,1,0,0,0]),Polrev([-2,5,6,-4,-2,1]),Polrev([-2,-3,6,-2,-2,1]),Polrev([-744,535,706,-347,-151,65]),Polrev([-102152,78787,91318,-50436,-18199,8693])], K);
magma: E := EllipticCurve([K![-2,0,1,0,0,0],K![-2,5,6,-4,-2,1],K![-2,-3,6,-2,-2,1],K![-744,535,706,-347,-151,65],K![-102152,78787,91318,-50436,-18199,8693]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^4-a^3-3a^2+a-2)\) | = | \((a^4-a^3-3a^2+a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 29 \) | = | \(29\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-7a^5+5a^4+39a^3-9a^2-56a+3)\) | = | \((a^4-a^3-3a^2+a-2)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 20511149 \) | = | \(29^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{34286919698837807988428}{20511149} a^{5} - \frac{11384698874034008601269}{20511149} a^{4} + \frac{179179369561896340242986}{20511149} a^{3} + \frac{74989014479850097421206}{20511149} a^{2} - \frac{168020705700556465895183}{20511149} a - \frac{14701659055200981175163}{20511149} \) | ||
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: | \( 3.9871607907008414956247136122427783107 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.20285 \) | ||
| Analytic order of Ш: | \( 625 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-a^3-3a^2+a-2)\) | \(29\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
29.3-c
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.