Properties

Label 5.5.81509.1-2.1-a8
Base field 5.5.81509.1
Conductor norm \( 2 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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Base field 5.5.81509.1

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 5 x^{3} + 3 x^{2} + 5 x - 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 5, 3, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-2, 5, 3, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 5, 3, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{4}-4a^{2}-a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(-5a^{4}+10a^{3}+11a^{2}-18a-8\right){x}-3a^{4}+a^{3}+18a^{2}-4a-21\)
sage: E = EllipticCurve([K([-1,-3,0,1,0]),K([1,-4,-1,1,0]),K([1,-1,-4,0,1]),K([-8,-18,11,10,-5]),K([-21,-4,18,1,-3])])
 
gp: E = ellinit([Polrev([-1,-3,0,1,0]),Polrev([1,-4,-1,1,0]),Polrev([1,-1,-4,0,1]),Polrev([-8,-18,11,10,-5]),Polrev([-21,-4,18,1,-3])], K);
 
magma: E := EllipticCurve([K![-1,-3,0,1,0],K![1,-4,-1,1,0],K![1,-1,-4,0,1],K![-8,-18,11,10,-5],K![-21,-4,18,1,-3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-2)\) = \((a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2 \) = \(2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^3+a^2+3a+2)\) = \((a^2-2)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 256 \) = \(2^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1379657237}{256} a^{4} - \frac{1096285751}{256} a^{3} - \frac{7471095507}{256} a^{2} + \frac{2559257005}{256} a + \frac{8910195159}{256} \)
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: \(\Z/2\Z\oplus\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{7}{4} a^{2} + \frac{3}{4} a - \frac{11}{4} : -\frac{7}{8} a^{4} + \frac{5}{8} a^{3} + \frac{33}{8} a^{2} - \frac{7}{8} a - \frac{21}{8} : 1\right)$ $\left(-3 a^{3} + 5 a^{2} + 6 a - 5 : a^{4} - 11 a^{3} + 13 a^{2} + 20 a - 11 : 1\right)$
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: \( 3552.1368101567653393770755466614061443 \)
Tamagawa product: \( 8 \)
Torsion order: \(8\)
Leading coefficient: \( 1.55523820 \)
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\((j)_{-}\)
\((a^2-2)\) \(2\) \(8\) \(I_{8}\) 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
\(3\) 3B

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 2.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.