Properties

Label 4.4.1125.1-31.2-a3
Base field \(\Q(\zeta_{15})^+\)
Conductor \((-a^3+5a-2)\)
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

\({y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{3}-3a\right){x}^{2}+\left(15a^{3}-5a^{2}-25a-31\right){x}-47a^{3}+4a^{2}+161a-22\)
sage: E = EllipticCurve([K([1,1,0,0]),K([0,-3,0,1]),K([-1,0,1,0]),K([-31,-25,-5,15]),K([-22,161,4,-47])])
 
gp: E = ellinit([Pol(Vecrev([1,1,0,0])),Pol(Vecrev([0,-3,0,1])),Pol(Vecrev([-1,0,1,0])),Pol(Vecrev([-31,-25,-5,15])),Pol(Vecrev([-22,161,4,-47]))], K);
 
magma: E := EllipticCurve([K![1,1,0,0],K![0,-3,0,1],K![-1,0,1,0],K![-31,-25,-5,15],K![-22,161,4,-47]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+5a-2)\) = \((-a^3+5a-2)\)
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: \((58a^3-278a^2-255a+1713)\) = \((-a^3+5a-2)^{8}\)
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: \( -\frac{96230619458397643875589}{852891037441} a^{3} - \frac{32551084210297485030913}{852891037441} a^{2} + \frac{341360629620385537992211}{852891037441} a + \frac{71907211552629246852068}{852891037441} \)
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: $\left(2 a^{3} - 5 a - 5 : -2 a^{3} - 2 a^{2} + 9 a + 4 : 1\right)$ $\left(-a^{3} - \frac{1}{4} a^{2} + \frac{1}{2} a + \frac{23}{4} : \frac{9}{8} a^{3} + \frac{11}{8} a^{2} - \frac{41}{8} a - \frac{23}{8} : 1\right)$
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: \( 226.240248527035 \)
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\((j)_{-}\)
\((-a^3+5a-2)\) \(31\) \(2\) \(I_{8}\) 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 31.2-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.