Properties

Label 2.0.3.1-61731.2-b1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 61731 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(11237a-5950\right){x}+322105a+82626\)
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([1,1]),K([-5950,11237]),K([82626,322105])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([1,1]),Polrev([-5950,11237]),Polrev([82626,322105])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,1],K![1,1],K![-5950,11237],K![82626,322105]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-285a+171)\) = \((-2a+1)^{2}\cdot(-5a+3)^{2}\cdot(-5a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 61731 \) = \(3^{2}\cdot19^{2}\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-174344218614a+866931329547)\) = \((-2a+1)^{7}\cdot(-5a+3)^{14}\cdot(-5a+2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 630821371472387493718347 \) = \(3^{7}\cdot19^{14}\cdot19^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{7240152655469734}{50950689123} a - \frac{5512832666599067}{50950689123} \)
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: $\left(41 a - \frac{249}{4} : -21 a + \frac{245}{8} : 1\right)$
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: \( 0.10810465534732925379326435365729001578 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 0.99862802984691617642153269630275646570 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(4\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((-5a+3)\) \(19\) \(4\) \(I_{8}^{*}\) Additive \(-1\) \(2\) \(14\) \(8\)
\((-5a+2)\) \(19\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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 and 8.
Its isogeny class 61731.2-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.