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([-4,4,5,-5,-1,1]),K([6,-5,-9,5,2,-1]),K([-1,1,0,-4,0,1]),K([31,-25,-86,48,35,-18]),K([14,64,-169,-32,152,-51])])
gp: E = ellinit([Polrev([-4,4,5,-5,-1,1]),Polrev([6,-5,-9,5,2,-1]),Polrev([-1,1,0,-4,0,1]),Polrev([31,-25,-86,48,35,-18]),Polrev([14,64,-169,-32,152,-51])], K);
magma: E := EllipticCurve([K![-4,4,5,-5,-1,1],K![6,-5,-9,5,2,-1],K![-1,1,0,-4,0,1],K![31,-25,-86,48,35,-18],K![14,64,-169,-32,152,-51]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+5a^3-5a+1)\) | = | \((-a^5+5a^3-5a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((3a^5-2a^4-15a^3+6a^2+14a-1)\) | = | \((-a^5+5a^3-5a+1)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -961 \) | = | \(-31^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{139573028727470950776}{961} a^{5} + \frac{146091909064081505603}{961} a^{4} - \frac{398857975268896914941}{961} a^{3} - \frac{258052846331700559100}{961} a^{2} + \frac{5474415068525499730}{31} a + \frac{68193984513407074549}{961} \) | ||
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: | $\left(\frac{5}{2} a^{5} - 4 a^{4} - \frac{39}{4} a^{3} + 15 a^{2} + 4 a - \frac{27}{4} : 4 a^{5} - 6 a^{4} - \frac{151}{8} a^{3} + \frac{105}{4} a^{2} + \frac{49}{4} a - 13 : 1\right)$ |
| 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: | \( 2500.6071233083976628084739470829728988 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.62410 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+5a^3-5a+1)\) | \(31\) | \(2\) | \(I_{2}\) | Non-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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
31.1-b
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.