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([6,3/2,-1/2]),K([-5,1/2,1/2]),K([-815260204308,-1607741225073,-458920937794]),K([-9103362981097420174,-17952368917266193425,-5124405501726437361])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([6,3/2,-1/2]),Polrev([-5,1/2,1/2]),Polrev([-815260204308,-1607741225073,-458920937794]),Polrev([-9103362981097420174,-17952368917266193425,-5124405501726437361])], K);
magma: E := EllipticCurve([K![1,0,0],K![6,3/2,-1/2],K![-5,1/2,1/2],K![-815260204308,-1607741225073,-458920937794],K![-9103362981097420174,-17952368917266193425,-5124405501726437361]]);
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: | \((-864a^2+2144a+10624)\) | = | \((3/2a^2-5/2a-19)^{6}\cdot(1/2a^2-1/2a-8)^{24}\cdot(-a-3)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -68719476736 \) | = | \(-2^{6}\cdot2^{24}\cdot2^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{65893712942219869}{33554432} a^{2} - \frac{256210319209888449}{33554432} a - \frac{91258046794472241}{16777216} \) | ||
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 | $\left(\frac{7047909}{32} a^{2} + \frac{24690957}{32} a + \frac{3130083}{8} : \frac{52399698211}{128} a^{2} + \frac{183572262947}{128} a + \frac{23271649535}{32} : 1\right)$ |
| Height | \(1.5473998428562028741698930620017645970\) |
| 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: | $\left(\frac{1433541}{8} a^{2} + \frac{5022127}{8} a + \frac{636657}{2} : -\frac{1433545}{16} a^{2} - \frac{5022131}{16} a - \frac{636647}{4} : 1\right)$ |
| 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: | \( 1.5473998428562028741698930620017645970 \) | ||
| Period: | \( 6.0074166183622993828003360910991836199 \) | ||
| Tamagawa product: | \( 8 \) = \(2\cdot2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.2970989113337995914436815665669356380 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((3/2a^2-5/2a-19)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
| \((1/2a^2-1/2a-8)\) | \(2\) | \(2\) | \(I_{24}\) | Non-split multiplicative | \(1\) | \(1\) | \(24\) | \(24\) |
| \((-a-3)\) | \(2\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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, 4, 6 and 12.
Its isogeny class
8.1-d
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.