Properties

Label 3.3.316.1-64.7-a2
Base field 3.3.316.1
Conductor norm \( 64 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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(Polrev([2, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(1698582370190a^{2}-899087811257a-7217515206526\right){x}-1762034893335112832a^{2}+932674283807826223a+7487133894869004409\)
sage: E = EllipticCurve([K([-3,0,1]),K([4,-1,-1]),K([-3,0,1]),K([-7217515206526,-899087811257,1698582370190]),K([7487133894869004409,932674283807826223,-1762034893335112832])])
 
gp: E = ellinit([Polrev([-3,0,1]),Polrev([4,-1,-1]),Polrev([-3,0,1]),Polrev([-7217515206526,-899087811257,1698582370190]),Polrev([7487133894869004409,932674283807826223,-1762034893335112832])], K);
 
magma: E := EllipticCurve([K![-3,0,1],K![4,-1,-1],K![-3,0,1],K![-7217515206526,-899087811257,1698582370190],K![7487133894869004409,932674283807826223,-1762034893335112832]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^2-3a-1)\) = \((-a+1)^{6}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 64 \) = \(2^{6}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-76a^2+105a+249)\) = \((-a+1)^{22}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -4194304 \) = \(-2^{22}\)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{1463291}{4} a^{2} + \frac{387273}{2} a + \frac{6217729}{4} : -\frac{4065697}{8} a^{2} + \frac{538009}{2} a + \frac{17275709}{8} : 1\right)$
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: \( 57.283066795226100268970131976893757616 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 1.6112121344811538096296176438321353910 \)
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\((j)_{-}\)
\((-a+1)\) \(2\) \(2\) \(I_{12}^{*}\) Additive \(-1\) \(6\) \(22\) \(4\)

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 64.7-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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

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