Properties

Label 2.2.21.1-1156.1-e1
Base field \(\Q(\sqrt{21}) \)
Conductor \((34)\)
Conductor norm \( 1156 \)
CM no
Base change yes: 34.a4,14994.u4
Q-curve yes
Torsion order \( 6 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{21}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 5 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 5)
 
gp: K = nfinit(a^2 - a - 5);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\(y^2+xy=x^{3}-3x+1\)
sage: E = EllipticCurve(K, [1, 0, 0, -3, 1])
 
gp: E = ellinit([1, 0, 0, -3, 1],K)
 
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -3, 1]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((34)\) = \( \left(2\right) \cdot \left(-2 a + 3\right) \cdot \left(2 a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 1156 \) = \( 4 \cdot 17^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((1088)\) = \( \left(2\right)^{6} \cdot \left(-2 a + 3\right) \cdot \left(2 a + 1\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1183744 \) = \( 4^{6} \cdot 17^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3048625}{1088} \)
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 $\left(-\frac{4}{25} a - \frac{26}{25} : \frac{8}{125} a + \frac{277}{125} : 1\right)$
Height \(1.55380276178178\)
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: $\left(0 : 1 : 1\right)$
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: \( 1.55380276178178 \)
Period: \( 20.2109887435620 \)
Tamagawa product: \( 6 \)  =  \(1\cdot1\cdot( 2 \cdot 3 )\)
Torsion order: \(6\)
Leading coefficient: \(2.28429688298469\)
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\((j)_{-}\)
\( \left(-2 a + 3\right) \) \(17\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(2 a + 1\right) \) \(17\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(2\right) \) \(4\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)

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

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-e consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base change of elliptic curves 34.a4, 14994.u4, defined over \(\Q\), so it is also a \(\Q\)-curve.