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([1,0,0,0,0,0]),K([0,2,-4,-1,1,0]),K([0,2,0,-4,0,1]),K([-12,8,25,-13,-6,3]),K([16,-15,-30,22,7,-5])])
gp: E = ellinit([Polrev([1,0,0,0,0,0]),Polrev([0,2,-4,-1,1,0]),Polrev([0,2,0,-4,0,1]),Polrev([-12,8,25,-13,-6,3]),Polrev([16,-15,-30,22,7,-5])], K);
magma: E := EllipticCurve([K![1,0,0,0,0,0],K![0,2,-4,-1,1,0],K![0,2,0,-4,0,1],K![-12,8,25,-13,-6,3],K![16,-15,-30,22,7,-5]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+2a^4+5a^3-8a^2-4a+3)\) | = | \((-a^5+2a^4+5a^3-8a^2-4a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 73 \) | = | \(73\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^4-a^3-4a^2+3a-7)\) | = | \((-a^5+2a^4+5a^3-8a^2-4a+3)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 389017 \) | = | \(73^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{896212872}{389017} a^{5} - \frac{2330492990}{389017} a^{4} - \frac{1852149899}{389017} a^{3} + \frac{8270889498}{389017} a^{2} - \frac{4711142963}{389017} a + \frac{266102902}{389017} \) | ||
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 | $\left(a^{4} - a^{3} - 5 a^{2} + 3 a + 6 : a^{5} + a^{4} - 7 a^{3} - 6 a^{2} + 10 a + 8 : 1\right)$ |
| Height | \(0.045430237162172782198382435089503717497\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(1 : -a^{5} + a^{4} + 4 a^{3} - 4 a^{2} - 2 a + 2 : 1\right)$ |
| 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: | \( 0.045430237162172782198382435089503717497 \) | ||
| Period: | \( 89096.364666198959907299665524573233413 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.62889 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+2a^4+5a^3-8a^2-4a+3)\) | \(73\) | \(3\) | \(I_{3}\) | 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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
73.2-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.