Properties

Label 2.0.3.1-130816.3-a1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 130816 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

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

Weierstrass equation

\({y}^2={x}^{3}-a{x}^{2}+\left(-181a+158\right){x}+177a-1031\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([158,-181]),K([-1031,177])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([158,-181]),Polrev([-1031,177])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![158,-181],K![-1031,177]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((400a-304)\) = \((2)^{4}\cdot(3a-2)\cdot(-9a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 130816 \) = \(4^{4}\cdot7\cdot73\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2826240a-30613504)\) = \((2)^{13}\cdot(3a-2)^{4}\cdot(-9a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 858653150150656 \) = \(4^{13}\cdot7^{4}\cdot73^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{146913674955}{25589858} a - \frac{269643786736}{12794929} \)
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 $\left(15 a - 1 : 42 a - 28 : 1\right)$
Height \(0.99988076897644076717052414562393604889\)
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: $\left(7 a + 2 : 0 : 1\right)$
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: \( 0.99988076897644076717052414562393604889 \)
Period: \( 0.68129390697308437131846803773951298526 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 3.1463865730791940537137401563334620943 \)
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\((j)_{-}\)
\((2)\) \(4\) \(2\) \(I_{5}^{*}\) Additive \(-1\) \(4\) \(13\) \(1\)
\((3a-2)\) \(7\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-9a+1)\) \(73\) \(2\) \(I_{2}\) 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.
Its isogeny class 130816.3-a 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.