Properties

Label 2.0.3.1-71148.2-g4
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-308a+154)\)
Conductor norm \( 71148 \)
CM no
Base change yes: 462.f1,1386.c1
Q-curve yes
Torsion order \( 6 \)
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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-14133a+14133\right){x}-647829\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([14133,-14133]),K([-647829,0])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([14133,-14133])),Pol(Vecrev([-647829,0]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![14133,-14133],K![-647829,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-308a+154)\) = \((-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(11)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 71148 \) = \(3\cdot4\cdot7\cdot7\cdot121\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((25367712678)\) = \((-2a+1)^{8}\cdot(2)\cdot(-3a+1)^{6}\cdot(3a-2)^{6}\cdot(11)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 643520846513561931684 \) = \(3^{8}\cdot4\cdot7^{6}\cdot7^{6}\cdot121^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{312196988566716625}{25367712678} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(21 a : -714 a + 357 : 1\right)$
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: \( 0.124541854795669 \)
Tamagawa product: \( 864 \)  =  \(2^{3}\cdot1\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot3\)
Torsion order: \(6\)
Leading coefficient: \( 3.45140512279942 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((2)\) \(4\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-3a+1)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((3a-2)\) \(7\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((11)\) \(121\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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
\(3\) 3B.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 71148.2-g consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base change of 462.f1, 1386.c1, defined over \(\Q\), so it is also a \(\Q\)-curve.