Properties

Label 2.0.3.1-134199.2-a2
Base field \(\Q(\sqrt{-3}) \)
Conductor \((402a-315)\)
Conductor norm \( 134199 \)
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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(384a+242\right){x}-3266a+5966\)
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([0,1]),K([242,384]),K([5966,-3266])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([242,384])),Pol(Vecrev([5966,-3266]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,-1],K![0,1],K![242,384],K![5966,-3266]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((402a-315)\) = \((-2a+1)^{2}\cdot(-4a+1)\cdot(-6a+1)\cdot(-7a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 134199 \) = \(3^{2}\cdot13\cdot31\cdot37\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((270545022a-116488611)\) = \((-2a+1)^{10}\cdot(-4a+1)^{8}\cdot(-6a+1)\cdot(-7a+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 55248791595945363 \) = \(3^{10}\cdot13^{8}\cdot31\cdot37\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{38041716070089006}{935643136987} a - \frac{407649753979711613}{8420788232883} \)
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 $\left(15 a - 11 : -8 a + 1 : 1\right)$
Height \(1.18140000486658\)
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: $\left(15 a - \frac{53}{4} : -8 a + \frac{53}{8} : 1\right)$
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: \( 1.18140000486658 \)
Period: \( 0.435926802047762 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\cdot1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 2.37870124275888 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(4\) \(I_{4}^{*}\) Additive \(-1\) \(2\) \(10\) \(4\)
\((-4a+1)\) \(13\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((-6a+1)\) \(31\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-7a+3)\) \(37\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 and 4.
Its isogeny class 134199.2-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.