Properties

Label 4.4.1125.1-31.4-b4
Base field \(\Q(\zeta_{15})^+\)
Conductor \((2a^3-8a+1)\)
Conductor norm \( 31 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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

\({y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}={x}^{3}+\left(a^{3}+a^{2}-3a-3\right){x}^{2}+\left(-20a^{3}-20a^{2}+55a+6\right){x}-92a^{3}-85a^{2}+241a+58\)
sage: E = EllipticCurve([K([-2,-3,1,1]),K([-3,-3,1,1]),K([0,0,0,0]),K([6,55,-20,-20]),K([58,241,-85,-92])])
 
gp: E = ellinit([Pol(Vecrev([-2,-3,1,1])),Pol(Vecrev([-3,-3,1,1])),Pol(Vecrev([0,0,0,0])),Pol(Vecrev([6,55,-20,-20])),Pol(Vecrev([58,241,-85,-92]))], K);
 
magma: E := EllipticCurve([K![-2,-3,1,1],K![-3,-3,1,1],K![0,0,0,0],K![6,55,-20,-20],K![58,241,-85,-92]]);
 

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: \((-336a^3-81a^2+1286a+983)\) = \((2a^3-8a+1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 852891037441 \) = \(31^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{63679535248100158844676}{852891037441} a^{3} + \frac{52668771245192606365444}{852891037441} a^{2} - \frac{158487521534002991503115}{852891037441} a - \frac{34852964110250777095970}{852891037441} \)
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: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(2 a^{3} + 2 a^{2} - 6 a + 2 : -3 a^{3} - 3 a^{2} + 8 a + 4 : 1\right)$ $\left(-a^{3} - a^{2} + \frac{11}{4} a - \frac{3}{2} : 2 a^{3} + \frac{15}{8} a^{2} - \frac{11}{2} a - \frac{21}{8} : 1\right)$
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: \( 47.3508490162373 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 0.705864781284238 \)
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\((j)_{-}\)
\((2a^3-8a+1)\) \(31\) \(2\) \(I_{8}\) Non-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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 31.4-b 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.