Properties

Label 2.0.11.1-24336.2-b1
Base field \(\Q(\sqrt{-11}) \)
Conductor norm \( 24336 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{-11}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
 
gp: K = nfinit(Polrev([3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+{x}^{2}-13{x}-4\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-13,0]),K([-4,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,0]),Polrev([-13,0]),Polrev([-4,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-13,0],K![-4,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((156)\) = \((-a)\cdot(a-1)\cdot(2)^{2}\cdot(13)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 24336 \) = \(3\cdot3\cdot4^{2}\cdot169\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((151632)\) = \((-a)^{6}\cdot(a-1)^{6}\cdot(2)^{4}\cdot(13)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 22992263424 \) = \(3^{6}\cdot3^{6}\cdot4^{4}\cdot169\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{16384000}{9477} \)
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: $\left(-1 : 3 : 1\right)$
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: \( 1.8829553242367890425802423755885015396 \)
Tamagawa product: \( 108 \)  =  \(( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot3\cdot1\)
Torsion order: \(6\)
Leading coefficient: \( 3.4063943495429587673159517280943793245 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((a-1)\) \(3\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((2)\) \(4\) \(3\) \(IV\) Additive \(1\) \(2\) \(4\) \(0\)
\((13)\) \(169\) \(1\) \(I_{1}\) 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
\(3\) 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 24336.2-b consists of curves linked by isogenies of degrees dividing 6.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 156.b4
\(\Q\) 18876.k4