Base field 4.4.15952.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -6, 0, 1]))
gp: K = nfinit(Polrev([1, -2, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-6,0,1]),K([0,-10,-1,2]),K([-2,0,1,0]),K([-5,-46,-2,8]),K([25,12,-4,-2])])
gp: E = ellinit([Polrev([-1,-6,0,1]),Polrev([0,-10,-1,2]),Polrev([-2,0,1,0]),Polrev([-5,-46,-2,8]),Polrev([25,12,-4,-2])], K);
magma: E := EllipticCurve([K![-1,-6,0,1],K![0,-10,-1,2],K![-2,0,1,0],K![-5,-46,-2,8],K![25,12,-4,-2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+6a+4)\) | = | \((a^2+2a)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^3-2a^2-12a-3)\) | = | \((a^2+2a)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2187 \) | = | \(3^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{40153079888}{3} a^{3} - \frac{11056865288}{3} a^{2} + \frac{237872824648}{3} a + \frac{145810516136}{3} \) | ||
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 | $\left(-a^{3} + 5 a + 3 : -a^{3} + 6 a : 1\right)$ |
| Height | \(0.32363630688438263996662183817381777955\) |
| Torsion structure: | \(\Z/4\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{1}{2} a^{3} + \frac{5}{2} a + 2 : \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - 2 a + 5 : 1\right)$ |
| 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: | \( 0.32363630688438263996662183817381777955 \) | ||
| Period: | \( 902.17265257699588509542637861501447047 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.31174179264793 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2+2a)\) | \(3\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
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, 4, 6 and 12.
Its isogeny class
9.1-a
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.