Base field 3.3.1620.1
Generator \(a\), with minimal polynomial \( x^{3} - 12 x - 14 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -12, 0, 1]))
gp: K = nfinit(Polrev([-14, -12, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -12, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-8,-1,1]),K([-9,-2,1]),K([-7,-1,1]),K([-52281,-7206,5189]),K([-3632976,-502080,360950])])
gp: E = ellinit([Polrev([-8,-1,1]),Polrev([-9,-2,1]),Polrev([-7,-1,1]),Polrev([-52281,-7206,5189]),Polrev([-3632976,-502080,360950])], K);
magma: E := EllipticCurve([K![-8,-1,1],K![-9,-2,1],K![-7,-1,1],K![-52281,-7206,5189],K![-3632976,-502080,360950]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-3a-3)\) | = | \((a+3)\cdot(2a^2-3a-19)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(5\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-47500a^2-1520000a-1573125)\) | = | \((a+3)^{9}\cdot(2a^2-3a-19)^{18}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -7450580596923828125 \) | = | \(-5^{9}\cdot5^{18}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{757976709141480384}{3814697265625} a^{2} + \frac{1010791944377174016}{3814697265625} a + \frac{7806263645751303024}{3814697265625} \) | ||
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 | $\left(-\frac{1739}{49} a^{2} + \frac{2329}{49} a + \frac{2550}{7} : \frac{56146}{343} a^{2} - \frac{75944}{343} a - \frac{82067}{49} : 1\right)$ |
| Height | \(2.1943021878809504055863088329650305134\) |
| 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: | $\left(19 a^{2} - 29 a - 180 : 22 a^{2} - 18 a - 273 : 1\right)$ |
| 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: | \( 2.1943021878809504055863088329650305134 \) | ||
| Period: | \( 4.5465983204133118357427823726644381094 \) | ||
| Tamagawa product: | \( 162 \) = \(3^{2}\cdot( 2 \cdot 3^{2} )\) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 3.3462569370500240681823464047088695097 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(5\) | \(9\) | \(I_{9}\) | Split multiplicative | \(-1\) | \(1\) | \(9\) | \(9\) |
| \((2a^2-3a-19)\) | \(5\) | \(18\) | \(I_{18}\) | Split multiplicative | \(-1\) | \(1\) | \(18\) | \(18\) |
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
25.1-f
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.