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([-1,-1,0]),K([0,1,0]),K([-3660,-5257/2,1529/2]),K([-15043,38357/2,-7035/2])])
gp: E = ellinit([Polrev([1,0,0]),Polrev([-1,-1,0]),Polrev([0,1,0]),Polrev([-3660,-5257/2,1529/2]),Polrev([-15043,38357/2,-7035/2])], K);
magma: E := EllipticCurve([K![1,0,0],K![-1,-1,0],K![0,1,0],K![-3660,-5257/2,1529/2],K![-15043,38357/2,-7035/2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((1)\) | = | \((1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 1 \) | = | 1 |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((1)\) | = | \((1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -1 \) | = | -1 |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -57005 a^{2} + 108767 a + 791289 \) | ||
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 | $\left(\frac{1049}{32} a^{2} - \frac{3917}{32} a - \frac{1789}{16} : \frac{88911}{128} a^{2} - \frac{345947}{128} a - \frac{122795}{64} : 1\right)$ |
| Height | \(0.97294313207849206986455156571462942670\) |
| 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: | $\left(\frac{11}{4} a^{2} - \frac{11}{2} a - \frac{117}{4} : -\frac{11}{8} a^{2} + \frac{9}{4} a + \frac{117}{8} : 1\right)$ |
| 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: | \( 0.97294313207849206986455156571462942670 \) | ||
| Period: | \( 183.09895535535788136690866122701170210 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.1071779843675910361723903410918946459 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
No primes of bad reduction.
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 4.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.