Properties

Label 2.0.3.1-123627.2-b5
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-406 a + 203)\)
Conductor norm \( 123627 \)
CM no
Base change yes: 1827.d4,609.a4
Q-curve yes
Torsion order \( 8 \)
Rank \( 2 \)

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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

\(y^2+axy+y=x^{3}+\left(a-1\right)x^{2}+12788ax+551346\)
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([1,0]),K([0,12788]),K([551346,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,12788])),Pol(Vecrev([551346,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,1],K![1,0],K![0,12788],K![551346,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-406 a + 203)\) = \( \left(-2 a + 1\right) \cdot \left(-3 a + 1\right) \cdot \left(3 a - 2\right) \cdot \left(29\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 123627 \) = \( 3 \cdot 7^{2} \cdot 841 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1472026689)\) = \( \left(-2 a + 1\right)^{12} \cdot \left(-3 a + 1\right)^{4} \cdot \left(3 a - 2\right)^{4} \cdot \left(29\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2166862573128302721 \) = \( 3^{12} \cdot 7^{8} \cdot 841^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{231331938231569617}{1472026689} \)
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: \(2\)
Generators $\left(\frac{196}{3} a - 66 : -\frac{82}{9} a + \frac{233}{9} : 1\right)$ $\left(\frac{323}{4} a - \frac{323}{4} : \frac{2167}{8} : 1\right)$
Heights \(2.71692228260522\) \(3.51346023908212\)
Torsion structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(72 a - 72 : -66 : 1\right)$ $\left(\frac{259}{4} a - \frac{259}{4} : \frac{255}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 6.45969769970693 \)
Period: \( 0.151347556869064 \)
Tamagawa product: \( 64 \)  =  \(2\cdot2^{2}\cdot2^{2}\cdot2\)
Torsion order: \(8\)
Leading coefficient: \(4.51561564217903\)
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\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(3\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\( \left(-3 a + 1\right) \) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(3 a - 2\right) \) \(7\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(29\right) \) \(841\) \(2\) \(I_{2}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 123627.2-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 1827.d4, 609.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.