Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,4,5,-5,-1,1]),K([-3,5,11,-9,-3,2]),K([-6,5,9,-9,-2,2]),K([167,192,-1083,503,288,-150]),K([-1163,-2425,10175,-4519,-2764,1413])])
gp: E = ellinit([Polrev([-3,4,5,-5,-1,1]),Polrev([-3,5,11,-9,-3,2]),Polrev([-6,5,9,-9,-2,2]),Polrev([167,192,-1083,503,288,-150]),Polrev([-1163,-2425,10175,-4519,-2764,1413])], K);
magma: E := EllipticCurve([K![-3,4,5,-5,-1,1],K![-3,5,11,-9,-3,2],K![-6,5,9,-9,-2,2],K![167,192,-1083,503,288,-150],K![-1163,-2425,10175,-4519,-2764,1413]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-a^4-4a^3+4a^2+3a-1)\) | = | \((a^4-4a^2+a+2)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 49 \) | = | \(7^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-5a^5+12a^4+16a^3-50a^2+25)\) | = | \((a^4-4a^2+a+2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -5764801 \) | = | \(-7^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -21437955273398086628814342060 a^{5} - 8296492504833651562385453152 a^{4} + 95682539786756414754531486072 a^{3} + 46959878613528243860092427239 a^{2} - 42056416375390440919728940937 a - 15456346448131269056429661858 \) | ||
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: | \( 1081.5677609031029303584856238161150981 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.40492 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^4-4a^2+a+2)\) | \(7\) | \(1\) | \(IV^{*}\) | Additive | \(1\) | \(2\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.6.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
49.2-b
consists of curves linked by isogenies of
degree 7.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.