Properties

Label 2.0.11.1-27225.5-b3
Base field \(\Q(\sqrt{-11}) \)
Conductor \((165)\)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 2 \)

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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(Pol(Vecrev([3, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-24a+40\right){x}-167a-217\)
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([1,1]),K([40,-24]),K([-217,-167])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,1])),Pol(Vecrev([40,-24])),Pol(Vecrev([-217,-167]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![1,1],K![40,-24],K![-217,-167]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((165)\) = \((-a)\cdot(a-1)\cdot(-a-1)\cdot(a-2)\cdot(-2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27225 \) = \(3\cdot3\cdot5\cdot5\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-39830175a+33241725)\) = \((-a)^{6}\cdot(a-1)^{2}\cdot(-a-1)^{2}\cdot(a-2)^{6}\cdot(-2a+1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4540317078515625 \) = \(3^{6}\cdot3^{2}\cdot5^{2}\cdot5^{6}\cdot11^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1217478647}{11390625} a - \frac{534233498}{11390625} \)
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 $\left(3 : 9 a - 17 : 1\right)$ $\left(11 a - 19 : -13 a - 83 : 1\right)$
Heights \(0.886854045662583\) \(1.01959978674625\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-5 a : 2 a - 8 : 1\right)$ $\left(3 a - 4 : 4 : 1\right)$
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: \( 0.762066362073900 \)
Period: \( 0.690501211399760 \)
Tamagawa product: \( 64 \)  =  \(2\cdot2\cdot2\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 5.07704336248562 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((a-1)\) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-a-1)\) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-2)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-2a+1)\) \(11\) \(4\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 27225.5-b consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.