Base field 3.3.1849.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 14 x - 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -14, -1, 1]))
gp: K = nfinit(Polrev([-8, -14, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -14, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0,0]),K([-4,-1/2,1/2]),K([-5,1/2,1/2]),K([-118695812,-521078557/2,-219683109/2]),K([-4056508189528,-15678187411863/2,-4044776198393/2])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([-4,-1/2,1/2]),Polrev([-5,1/2,1/2]),Polrev([-118695812,-521078557/2,-219683109/2]),Polrev([-4056508189528,-15678187411863/2,-4044776198393/2])], K);
magma: E := EllipticCurve([K![1,0,0],K![-4,-1/2,1/2],K![-5,1/2,1/2],K![-118695812,-521078557/2,-219683109/2],K![-4056508189528,-15678187411863/2,-4044776198393/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((3/2a^2-5/2a-19)\cdot(1/2a^2-1/2a-8)\cdot(-a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(2\cdot2\cdot2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-40a^2+88a+96)\) | = | \((3/2a^2-5/2a-19)^{4}\cdot(1/2a^2-1/2a-8)^{10}\cdot(-a-3)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4194304 \) | = | \(2^{4}\cdot2^{10}\cdot2^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{4890658379189}{2048} a^{2} + \frac{18406527088015}{2048} a + \frac{6182502630031}{1024} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{1327}{2} a^{2} + \frac{21607}{2} a + 6395 : -332 a^{2} - 5402 a - 3195 : 1\right)$ | $\left(-\frac{1715}{8} a^{2} - \frac{46875}{8} a - \frac{14087}{4} : \frac{1711}{16} a^{2} + \frac{46871}{16} a + \frac{14107}{8} : 1\right)$ |
| 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: | \( 8.0223136799968813808709016468182252577 \) | ||
| Tamagawa product: | \( 32 \) = \(2\cdot2\cdot2^{3}\) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.4925234753482570010922607715010651642 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3/2a^2-5/2a-19)\) | \(2\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((1/2a^2-1/2a-8)\) | \(2\) | \(2\) | \(I_{10}\) | Non-split multiplicative | \(1\) | \(1\) | \(10\) | \(10\) |
| \((-a-3)\) | \(2\) | \(8\) | \(I_{8}\) | Split multiplicative | \(-1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
8.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.