Properties

Label 3.3.316.1-2.2-a4
Base field 3.3.316.1
Conductor \((-a+1)\)
Conductor norm \( 2 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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Base field 3.3.316.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, -4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(469933301874402734133a^{2}-248743488236812365957a-1996812643211595590900\right){x}-8108483751005890586477675293469a^{2}+4291954889112701686585656294515a+34454087009163876680154219600725\)
sage: E = EllipticCurve([K([-3,1,1]),K([2,0,-1]),K([0,1,0]),K([-1996812643211595590900,-248743488236812365957,469933301874402734133]),K([34454087009163876680154219600725,4291954889112701686585656294515,-8108483751005890586477675293469])])
 
gp: E = ellinit([Pol(Vecrev([-3,1,1])),Pol(Vecrev([2,0,-1])),Pol(Vecrev([0,1,0])),Pol(Vecrev([-1996812643211595590900,-248743488236812365957,469933301874402734133])),Pol(Vecrev([34454087009163876680154219600725,4291954889112701686585656294515,-8108483751005890586477675293469]))], K);
 
magma: E := EllipticCurve([K![-3,1,1],K![2,0,-1],K![0,1,0],K![-1996812643211595590900,-248743488236812365957,469933301874402734133],K![34454087009163876680154219600725,4291954889112701686585656294515,-8108483751005890586477675293469]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a+1)\) = \((-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2 \) = \(2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^2+a+7)\) = \((-a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -16 \) = \(2^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{32251567931279}{16} a^{2} + \frac{21655756210331}{8} a - \frac{13765401110581}{8} \)
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: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-6188258995 a^{2} + 3275548088 a + 26294782156 : -36997829715248 a^{2} + 19583564697064 a + 157208978065875 : 1\right)$
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: \( 171.477293083240 \)
Tamagawa product: \( 4 \)
Torsion order: \(8\)
Leading coefficient: \( 0.602896961660936 \)
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\((j)_{-}\)
\((-a+1)\) \(2\) \(4\) \(I_{4}\) 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, 8 and 16.
Its isogeny class 2.2-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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