Properties

Label 4.4.1125.1-31.4-a6
Base field \(\Q(\zeta_{15})^+\)
Conductor \((2a^3-8a+1)\)
Conductor norm \( 31 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{15})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -4, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, 4, -4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-2a\right){y}={x}^{3}+\left(-a^{3}+3a-1\right){x}^{2}+\left(59a^{3}-114a^{2}-239a+418\right){x}+45a^{3}+134a^{2}-121a-414\)
sage: E = EllipticCurve([K([-1,0,1,0]),K([-1,3,0,-1]),K([0,-2,0,1]),K([418,-239,-114,59]),K([-414,-121,134,45])])
 
gp: E = ellinit([Pol(Vecrev([-1,0,1,0])),Pol(Vecrev([-1,3,0,-1])),Pol(Vecrev([0,-2,0,1])),Pol(Vecrev([418,-239,-114,59])),Pol(Vecrev([-414,-121,134,45]))], K);
 
magma: E := EllipticCurve([K![-1,0,1,0],K![-1,3,0,-1],K![0,-2,0,1],K![418,-239,-114,59],K![-414,-121,134,45]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^3-8a+1)\) = \((2a^3-8a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 31 \) = \(31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-403593a^3-719940a^2+1212017a+1277583)\) = \((2a^3-8a+1)^{16}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 727423121747185263828481 \) = \(31^{16}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{102107545967007124835404235745}{727423121747185263828481} a^{3} - \frac{84890779633537182060967675121}{727423121747185263828481} a^{2} + \frac{253895730793338645242467504787}{727423121747185263828481} a + \frac{57110378827362524396580356024}{727423121747185263828481} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{13}{4} a^{3} - \frac{5}{2} a^{2} + 10 a + 3 : \frac{9}{4} a^{3} + \frac{9}{4} a^{2} - \frac{57}{8} a - \frac{11}{8} : 1\right)$
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: \( 14.1400155329397 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 0.843147624921494 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((2a^3-8a+1)\) \(31\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)

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, 4 and 8.
Its isogeny class 31.4-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.