Base field 6.6.434581.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,1,7,-6,-4,2]),K([-3,-2,11,0,-4,1]),K([0,2,7,-6,-4,2]),K([-1279,-90,3065,16,-2194,670]),K([-21619,-10467,73555,14638,-68427,20194])])
gp: E = ellinit([Polrev([-1,1,7,-6,-4,2]),Polrev([-3,-2,11,0,-4,1]),Polrev([0,2,7,-6,-4,2]),Polrev([-1279,-90,3065,16,-2194,670]),Polrev([-21619,-10467,73555,14638,-68427,20194])], K);
magma: E := EllipticCurve([K![-1,1,7,-6,-4,2],K![-3,-2,11,0,-4,1],K![0,2,7,-6,-4,2],K![-1279,-90,3065,16,-2194,670],K![-21619,-10467,73555,14638,-68427,20194]]);
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: | \( -299441694155033880148130654153024 a^{5} - 187917845986491657242090310571049 a^{4} + 704001225414123443403572247320027 a^{3} + 352597497846354900118604631201756 a^{2} - 271295440361411180696560081353475 a - 113961855027594415883614627224113 \) | ||
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: | \( 0.014802157632734448067338807327986183064 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.641303 \) | ||
| Analytic order of Ш: | \( 28561 \) (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 |
|---|---|
| \(3\) | 3B |
| \(13\) | 13B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 13 and 39.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 39.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.