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([-3,3,9,-7,-4,2]),K([-4,0,11,-5,-5,2]),K([-1,4,5,-4,-2,1]),K([-32,-2,72,-16,-30,10]),K([-26,18,25,-24,3,1])])
gp: E = ellinit([Polrev([-3,3,9,-7,-4,2]),Polrev([-4,0,11,-5,-5,2]),Polrev([-1,4,5,-4,-2,1]),Polrev([-32,-2,72,-16,-30,10]),Polrev([-26,18,25,-24,3,1])], K);
magma: E := EllipticCurve([K![-3,3,9,-7,-4,2],K![-4,0,11,-5,-5,2],K![-1,4,5,-4,-2,1],K![-32,-2,72,-16,-30,10],K![-26,18,25,-24,3,1]]);
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: | \((-a^4+2a^3+a^2-a+4)\) | = | \((a^3-3a)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -5041 \) | = | \(-71^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{58168686536343316256911}{5041} a^{5} + \frac{39830276056343603859101}{5041} a^{4} + \frac{285062017990100061908014}{5041} a^{3} + \frac{84087973968099499538051}{5041} a^{2} - \frac{122076980141719179710154}{5041} a - \frac{44225911390155224771855}{5041} \) | ||
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 | $\left(-a^{5} + 4 a^{4} - 12 a^{2} + 3 a + 9 : -4 a^{5} + 6 a^{4} + 18 a^{3} - 10 a^{2} - 17 a - 4 : 1\right)$ |
| Height | \(0.097983788799860610502864856391415823301\) |
| 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: | $\left(a^{5} - \frac{11}{4} a^{4} - \frac{11}{4} a^{3} + \frac{33}{4} a^{2} - \frac{7}{2} : \frac{5}{8} a^{5} - \frac{7}{4} a^{4} - \frac{5}{4} a^{3} + \frac{33}{8} a^{2} + \frac{3}{8} a - \frac{5}{2} : 1\right)$ |
| 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: | \( 0.097983788799860610502864856391415823301 \) | ||
| Period: | \( 5981.5737388344693552573314079832798314 \) | ||
| Tamagawa product: | \( 2 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-3a)\) | \(71\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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.