Base field 4.4.2525.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, 5, -4, -2, 1]))
gp: K = nfinit(Polrev([5, 5, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 5, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,-3,0,1]),K([3,3,0,-1]),K([-2,0,1,0]),K([-884,-1373,-80,304]),K([-16937,-26497,-1423,5992])])
gp: E = ellinit([Polrev([-3,-3,0,1]),Polrev([3,3,0,-1]),Polrev([-2,0,1,0]),Polrev([-884,-1373,-80,304]),Polrev([-16937,-26497,-1423,5992])], K);
magma: E := EllipticCurve([K![-3,-3,0,1],K![3,3,0,-1],K![-2,0,1,0],K![-884,-1373,-80,304],K![-16937,-26497,-1423,5992]]);
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: | \( -17038484321262621392677 a^{3} + 47324121859575160666349 a^{2} + 31360177851679811257113 a - 109574465986214263634831 \) | ||
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: | \( 0.42869629016433029105799593505762430899 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 0.418037376978190 \) | ||
| Analytic order of Ш: | \( 49 \) (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 |
|---|---|
| \(5\) | 5B.4.1[2] |
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5, 7 and 35.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 35.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.