Base field 6.6.434581.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,5,0,-5,-1,1]),K([3,-5,-12,10,6,-3]),K([-1,0,4,-3,-2,1]),K([2,-5,-4,8,3,-2]),K([2,0,-4,3,2,-1])])
gp: E = ellinit([Polrev([1,5,0,-5,-1,1]),Polrev([3,-5,-12,10,6,-3]),Polrev([-1,0,4,-3,-2,1]),Polrev([2,-5,-4,8,3,-2]),Polrev([2,0,-4,3,2,-1])], K);
magma: E := EllipticCurve([K![1,5,0,-5,-1,1],K![3,-5,-12,10,6,-3],K![-1,0,4,-3,-2,1],K![2,-5,-4,8,3,-2],K![2,0,-4,3,2,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-3a)\) | = | \((a^3-3a)\) |
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: | \((-a^5-6a^4+17a^3+28a^2-29a-21)\) | = | \((a^3-3a)^{3}\) |
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: | \( \frac{19503808031}{357911} a^{5} - \frac{53829939140}{357911} a^{4} - \frac{36653092470}{357911} a^{3} + \frac{126101211770}{357911} a^{2} - \frac{17962351604}{357911} a - \frac{25733383533}{357911} \) | ||
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: | \(1\) |
| Generator | $\left(6 a^{5} - 9 a^{4} - 27 a^{3} + 14 a^{2} + 25 a + 5 : 14 a^{5} - 18 a^{4} - 69 a^{3} + 21 a^{2} + 72 a + 21 : 1\right)$ |
| Height | \(0.016330631466643435083810809398569303883\) |
| 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: | $\left(a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} - 2 : a^{5} - a^{4} - 5 a^{3} + 4 a + 1 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.016330631466643435083810809398569303883 \) | ||
| Period: | \( 23926.294955337877421029325631933119325 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.66720 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a)\) | \(71\) | \(3\) | \(I_{3}\) | 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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
71.2-b
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.