Properties

Label 2.0.3.1-1156.1-a4
Base field \(\Q(\sqrt{-3}) \)
Conductor \((34)\)
Conductor norm \( 1156 \)
CM no
Base change yes: 34.a2,306.a2
Q-curve yes
Torsion order \( 6 \)
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

\(y^2+\left(a+1\right)xy=x^{3}-ax^{2}+\left(-103a+103\right)x-411\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([103,-103]),K([-411,0])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([103,-103])),Pol(Vecrev([-411,0]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![103,-103],K![-411,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((34)\) = \( \left(2\right) \cdot \left(17\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1156 \) = \( 4 \cdot 289 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((19652)\) = \( \left(2\right)^{2} \cdot \left(17\right)^{3} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 386201104 \) = \( 4^{2} \cdot 289^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{120920208625}{19652} \)
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 $\left(3 a : 10 a - 5 : 1\right)$
Height \(1.68788702948039\)
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: $\left(12 a : 22 a - 11 : 1\right)$
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: \( 1.68788702948039 \)
Period: \( 1.39678390661793 \)
Tamagawa product: \( 6 \)  =  \(2\cdot3\)
Torsion order: \(6\)
Leading coefficient: \(0.907445835755251\)
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\((j)_{-}\)
\( \left(2\right) \) \(4\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(17\right) \) \(289\) \(3\) \(I_{3}\) 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 1156.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base change of elliptic curves 34.a2, 306.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.