Properties

Label 2.0.3.1-12675.2-c1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 12675 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-130a+65)\) = \((-2a+1)\cdot(-4a+1)\cdot(4a-3)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 12675 \) = \(3\cdot13\cdot13\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-195)\) = \((-2a+1)^{2}\cdot(-4a+1)\cdot(4a-3)\cdot(5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 38025 \) = \(3^{2}\cdot13\cdot13\cdot25\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{4096}{195} \)
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 $\left(0 : -a : 1\right)$
Height \(0.099517256151416781914988041046347366291\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.099517256151416781914988041046347366291 \)
Period: \( 5.8192374973323256789526059405621657498 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.6748155243358071294342623805614749862 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-4a+1)\) \(13\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((4a-3)\) \(13\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((5)\) \(25\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 12675.2-c consists of this curve only.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 195.d1
\(\Q\) 585.a1