Properties

Label 3.3.316.1-32.1-b3
Base field 3.3.316.1
Conductor norm \( 32 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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(Polrev([2, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(122252877320a^{2}-64710474941a-519469656935\right){x}-34245221998777113a^{2}+18126563794145582a+145512761032249632\)
sage: E = EllipticCurve([K([0,1,0]),K([-3,-1,1]),K([0,1,0]),K([-519469656935,-64710474941,122252877320]),K([145512761032249632,18126563794145582,-34245221998777113])])
 
gp: E = ellinit([Polrev([0,1,0]),Polrev([-3,-1,1]),Polrev([0,1,0]),Polrev([-519469656935,-64710474941,122252877320]),Polrev([145512761032249632,18126563794145582,-34245221998777113])], K);
 
magma: E := EllipticCurve([K![0,1,0],K![-3,-1,1],K![0,1,0],K![-519469656935,-64710474941,122252877320],K![145512761032249632,18126563794145582,-34245221998777113]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^2+8)\) = \((a)^{4}\cdot(-a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 32 \) = \(2^{4}\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16a^2-64a+80)\) = \((a)^{8}\cdot(-a+1)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1048576 \) = \(2^{8}\cdot2^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{532560105997425}{4096} a^{2} - \frac{749207151764755}{2048} a + \frac{293646661918461}{2048} \)
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: $\left(-97753 a^{2} + 51746 a + 415372 : 7204920 a^{2} - 3813675 a - 30614702 : 1\right)$
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: \( 44.991602444967165512583781817947804505 \)
Tamagawa product: \( 12 \)  =  \(1\cdot( 2^{2} \cdot 3 )\)
Torsion order: \(4\)
Leading coefficient: \( 1.8982315332789852452688635903059982991 \)
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\((j)_{-}\)
\((a)\) \(2\) \(1\) \(I_0^{*}\) Additive \(1\) \(4\) \(8\) \(0\)
\((-a+1)\) \(2\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)

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

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 32.1-b consists of curves linked by isogenies of degrees dividing 12.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.