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([-7,-2,1]),K([-7,-1,1]),K([-8,-1,1]),K([-18,-8,2]),K([-24,-14,-1])])
gp: E = ellinit([Polrev([-7,-2,1]),Polrev([-7,-1,1]),Polrev([-8,-1,1]),Polrev([-18,-8,2]),Polrev([-24,-14,-1])], K);
magma: E := EllipticCurve([K![-7,-2,1],K![-7,-1,1],K![-8,-1,1],K![-18,-8,2],K![-24,-14,-1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^2+a+12)\) | = | \((a+2)\cdot(a+3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 10 \) | = | \(2\cdot5\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((128a^2-4672a-11008)\) | = | \((a+2)^{19}\cdot(a+3)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -204800000000 \) | = | \(-2^{19}\cdot5^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{18345049}{80000} a^{2} - \frac{1419129}{5000} a + \frac{7693911}{10000} \) | ||
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: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 6.6208123900816951025784859338347793770 \) | ||
| Tamagawa product: | \( 8 \) = \(1\cdot2^{3}\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.3159632507107133019710674161694664924 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+2)\) | \(2\) | \(1\) | \(I_{19}\) | Non-split multiplicative | \(1\) | \(1\) | \(19\) | \(19\) |
| \((a+3)\) | \(5\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 10.2-b consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.