Properties

Label 4.4.1125.1-31.3-a1
Base field \(\Q(\zeta_{15})^+\)
Conductor \((-2a^3-2a^2+6a+3)\)
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

\({y}^2+\left(a^{3}-2a\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-3a+2\right){x}^{2}+\left(495a^{3}+69a^{2}-1642a-353\right){x}-7129a^{3}-1543a^{2}+24030a+5034\)
sage: E = EllipticCurve([K([0,-2,0,1]),K([2,-3,-1,1]),K([-1,-3,1,1]),K([-353,-1642,69,495]),K([5034,24030,-1543,-7129])])
 
gp: E = ellinit([Pol(Vecrev([0,-2,0,1])),Pol(Vecrev([2,-3,-1,1])),Pol(Vecrev([-1,-3,1,1])),Pol(Vecrev([-353,-1642,69,495])),Pol(Vecrev([5034,24030,-1543,-7129]))], K);
 
magma: E := EllipticCurve([K![0,-2,0,1],K![2,-3,-1,1],K![-1,-3,1,1],K![-353,-1642,69,495],K![5034,24030,-1543,-7129]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^3-2a^2+6a+3)\) = \((-2a^3-2a^2+6a+3)\)
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: \((-25a^3+101a-11)\) = \((-2a^3-2a^2+6a+3)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 923521 \) = \(31^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{3352161032683862028986803952}{923521} a^{3} + \frac{9909977575470350951831216400}{923521} a^{2} - \frac{5978197446809240162656682401}{923521} a - \frac{1713525152117869864310632982}{923521} \)
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: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-5 a^{3} + 14 a + 4 : 32 a^{3} + 17 a^{2} - 114 a - 24 : 1\right)$
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: \( 226.240248527035 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 0.843147624921494 \)
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\((j)_{-}\)
\((-2a^3-2a^2+6a+3)\) \(31\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

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 31.3-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.