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Properties

Label 5.5.65657.1-32.1-a1
Base field 5.5.65657.1
Conductor norm \( 32 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 5.5.65657.1

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

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

Weierstrass equation

\({y}^2+\left(-a^{4}+a^{3}+5a^{2}-3a-3\right){x}{y}+\left(-2a^{4}+3a^{3}+9a^{2}-8a-7\right){y}={x}^{3}+\left(-a^{4}+a^{3}+5a^{2}-3a-3\right){x}^{2}+\left(-3a^{4}+4a^{3}+8a^{2}-5a-7\right){x}-a^{4}-2a^{3}+7a^{2}+a-6\)
sage: E = EllipticCurve([K([-3,-3,5,1,-1]),K([-3,-3,5,1,-1]),K([-7,-8,9,3,-2]),K([-7,-5,8,4,-3]),K([-6,1,7,-2,-1])])
 
gp: E = ellinit([Polrev([-3,-3,5,1,-1]),Polrev([-3,-3,5,1,-1]),Polrev([-7,-8,9,3,-2]),Polrev([-7,-5,8,4,-3]),Polrev([-6,1,7,-2,-1])], K);
 
magma: E := EllipticCurve([K![-3,-3,5,1,-1],K![-3,-3,5,1,-1],K![-7,-8,9,3,-2],K![-7,-5,8,4,-3],K![-6,1,7,-2,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 32 \) = \(32\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4)\) = \((2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1024 \) = \(-32^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 26869605 a^{4} - 40011890 a^{3} - \frac{497646017}{4} a^{2} + \frac{294534937}{2} a + \frac{162685211}{4} \)
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: \( 126.71418849573431515568760772776713868 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 0.989041978 \)
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\((j)_{-}\)
\((2)\) \(32\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 32.1-a consists of this curve only.

Base change

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

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