Properties

Label 3.3.316.1-8.1-a1
Base field 3.3.316.1
Conductor \((2)\)
Conductor norm \( 8 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
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

\({y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-a^{2}-8a+8\right){x}+5a^{2}-22a+11\)
sage: E = EllipticCurve([K([-2,0,1]),K([4,-1,-1]),K([-2,0,1]),K([8,-8,-1]),K([11,-22,5])])
 
gp: E = ellinit([Pol(Vecrev([-2,0,1])),Pol(Vecrev([4,-1,-1])),Pol(Vecrev([-2,0,1])),Pol(Vecrev([8,-8,-1])),Pol(Vecrev([11,-22,5]))], K);
 
magma: E := EllipticCurve([K![-2,0,1],K![4,-1,-1],K![-2,0,1],K![8,-8,-1],K![11,-22,5]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a)^{2}\cdot(-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 8 \) = \(2^{2}\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-6a^2+4a+20)\) = \((a)^{8}\cdot(-a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -512 \) = \(2^{8}\cdot2\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{33759}{2} a^{2} + 32803 a + 4371 \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(a^{2} - 3 a : 3 a^{2} - 8 a + 3 : 1\right)$
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: \( 145.938695996136 \)
Tamagawa product: \( 3 \)  =  \(3\cdot1\)
Torsion order: \(6\)
Leading coefficient: \( 0.684141088092391 \)
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\((j)_{-}\)
\((a)\) \(2\) \(3\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((-a+1)\) \(2\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 8.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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