Properties

Label 2.0.3.1-61731.3-b2
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 61731 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-3}) \)

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

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

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-128a+535\right){x}-4787a+13769\)
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([0,1]),K([535,-128]),K([13769,-4787])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([535,-128])),Pol(Vecrev([13769,-4787]))], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![0,1],K![535,-128],K![13769,-4787]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-285a+114)\) = \((-2a+1)^{2}\cdot(-5a+3)\cdot(-5a+2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 61731 \) = \(3^{2}\cdot19\cdot19^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((53202245040a-82452923811)\) = \((-2a+1)^{8}\cdot(-5a+3)^{4}\cdot(-5a+2)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5242282865421502717881 \) = \(3^{8}\cdot19^{4}\cdot19^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{67419143}{390963} \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{9}{4} a - 17 : \frac{23}{4} a + \frac{77}{8} : 1\right)$ $\left(-31 a + 21 : 20 a - 26 : 1\right)$
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: \( 0.216209310694659 \)
Tamagawa product: \( 64 \)  =  \(2^{2}\cdot2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 0.998628029846916 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(4\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)
\((-5a+3)\) \(19\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-5a+2)\) \(19\) \(4\) \(I_{4}^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 61731.3-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is a \(\Q\)-curve.

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