Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 7, 2, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 7, 2, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,3,1,-6,-1,1]),K([-9,-18,19,30,0,-4]),K([-6,-29,17,43,2,-6]),K([-75,-156,176,278,-3,-36]),K([1358,4623,-3654,-7802,-361,1119])])
gp: E = ellinit([Polrev([0,3,1,-6,-1,1]),Polrev([-9,-18,19,30,0,-4]),Polrev([-6,-29,17,43,2,-6]),Polrev([-75,-156,176,278,-3,-36]),Polrev([1358,4623,-3654,-7802,-361,1119])], K);
magma: E := EllipticCurve([K![0,3,1,-6,-1,1],K![-9,-18,19,30,0,-4],K![-6,-29,17,43,2,-6],K![-75,-156,176,278,-3,-36],K![1358,4623,-3654,-7802,-361,1119]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-7a^5+2a^4+50a^3+22a^2-30a-10)\) | = | \((-7a^5+2a^4+50a^3+22a^2-30a-10)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a^4-8a^3-25a^2+11a+12)\) | = | \((-7a^5+2a^4+50a^3+22a^2-30a-10)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -357911 \) | = | \(-71^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{235353321428599978}{357911} a^{5} + \frac{580741577725442818}{357911} a^{4} + \frac{1162430939297012415}{357911} a^{3} - \frac{1879545228122345414}{357911} a^{2} + \frac{264656034786687147}{357911} a + \frac{239407872830503392}{357911} \) | ||
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: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-13 a^{5} + 3 a^{4} + 92 a^{3} + 49 a^{2} - 55 a - 19 : 3 a^{5} + a^{4} - 25 a^{3} - 15 a^{2} + 19 a + 4 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 34.907982152988347661509028368625290501 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.29032 \) | ||
| Analytic order of Ш: | \( 81 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-7a^5+2a^4+50a^3+22a^2-30a-10)\) | \(71\) | \(1\) | \(I_{3}\) | Non-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 |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
71.5-c
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.