Properties

Label 2.0.3.1-61731.3-g1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 61731 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(238a-817\right){x}+3795a-8972\)
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([1,0]),K([-817,238]),K([-8972,3795])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,-1]),Polrev([1,0]),Polrev([-817,238]),Polrev([-8972,3795])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,-1],K![1,0],K![-817,238],K![-8972,3795]]);
 

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: \((-559859016a-668937693)\) = \((-2a+1)^{3}\cdot(-5a+3)^{4}\cdot(-5a+2)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1135430553480940593 \) = \(3^{3}\cdot19^{4}\cdot19^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5845799184}{130321} a - \frac{4133064549}{130321} \)
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: \(1\)
Generator $\left(-\frac{1}{4} a - \frac{77}{4} : 34 a - \frac{215}{8} : 1\right)$
Height \(2.7690401079907991677525174992602532079\)
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: $\left(a - \frac{61}{4} : -\frac{1}{2} a + \frac{57}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.7690401079907991677525174992602532079 \)
Period: \( 0.35616692849373169400489067692545008420 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 4.5552497921158811988318787349431652850 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(III\) Additive \(1\) \(2\) \(3\) \(0\)
\((-5a+3)\) \(19\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-5a+2)\) \(19\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)

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.
Its isogeny class 61731.3-g consists of curves linked by isogenies of degree 2.

Base change

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

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