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([-9,-28,23,44,1,-6]),K([6,12,-10,-21,-1,3]),K([2,3,-5,-7,0,1]),K([389,1326,-1411,-2223,-10,301]),K([3947,12811,-12957,-21196,-180,2885])])
gp: E = ellinit([Polrev([-9,-28,23,44,1,-6]),Polrev([6,12,-10,-21,-1,3]),Polrev([2,3,-5,-7,0,1]),Polrev([389,1326,-1411,-2223,-10,301]),Polrev([3947,12811,-12957,-21196,-180,2885])], K);
magma: E := EllipticCurve([K![-9,-28,23,44,1,-6],K![6,12,-10,-21,-1,3],K![2,3,-5,-7,0,1],K![389,1326,-1411,-2223,-10,301],K![3947,12811,-12957,-21196,-180,2885]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+a^4+7a^3-2a^2-7a)\) | = | \((-a^5+a^4+7a^3-2a^2-7a)\) |
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: | \((-80a^5+9a^4+556a^3+381a^2-236a-169)\) | = | \((-a^5+a^4+7a^3-2a^2-7a)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -128100283921 \) | = | \(-71^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{12188349191966910976545591421520911263}{128100283921} a^{5} + \frac{2795083691338226342388959981793334127}{128100283921} a^{4} + \frac{87472547643214485589591447553049289390}{128100283921} a^{3} + \frac{43036275357468336314681380908411365576}{128100283921} a^{2} - \frac{734527173722240035758902620347146720}{1804229351} a - \frac{15815145011644535445549863476994175471}{128100283921} \) | ||
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: | $\left(\frac{29}{4} a^{5} - 5 a^{4} - \frac{187}{4} a^{3} - \frac{23}{4} a^{2} + \frac{67}{4} a + \frac{1}{4} : -33 a^{5} + \frac{87}{8} a^{4} + \frac{1861}{8} a^{3} + \frac{777}{8} a^{2} - \frac{1095}{8} a - \frac{333}{8} : 1\right)$ |
| 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: | \( 17.453991076494173830754514184312645251 \) | ||
| Tamagawa product: | \( 2 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+a^4+7a^3-2a^2-7a)\) | \(71\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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
71.3-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.