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([0,0,0]),K([-8,-1,1]),K([-7,-2,1]),K([-6004,-6672,-1693]),K([-567289,-630109,-159791])])
gp: E = ellinit([Polrev([0,0,0]),Polrev([-8,-1,1]),Polrev([-7,-2,1]),Polrev([-6004,-6672,-1693]),Polrev([-567289,-630109,-159791])], K);
magma: E := EllipticCurve([K![0,0,0],K![-8,-1,1],K![-7,-2,1],K![-6004,-6672,-1693],K![-567289,-630109,-159791]]);
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: | \((125)\) | = | \((a+3)^{6}\cdot(2a^2-3a-19)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1953125 \) | = | \(5^{6}\cdot5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{183711891456}{125} \) | ||
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 | $\left(\frac{646}{49} a^{2} + \frac{445}{14} a + \frac{921}{49} : \frac{91853}{686} a^{2} + \frac{42209}{98} a + \frac{234043}{686} : 1\right)$ |
| Height | \(3.2914532818214256083794632494475457700\) |
| 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: | $\left(-5 a^{2} - \frac{29}{2} a - 6 : -\frac{1}{2} a^{2} + a + \frac{7}{2} : 1\right)$ |
| 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: | \( 3.2914532818214256083794632494475457700 \) | ||
| Period: | \( 3.0310655469422078904951882484429587396 \) | ||
| Tamagawa product: | \( 18 \) = \(( 2 \cdot 3 )\cdot3\) | ||
| Torsion order: | \(2\) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+3)\) | \(5\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((2a^2-3a-19)\) | \(5\) | \(3\) | \(I_{3}\) | 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.1.2 |
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 the base change of the following elliptic curve:
| Base field | Curve |
|---|---|
| \(\Q\) | 1620.f1 |