Properties

Label 2.0.11.1-27225.5-d7
Base field \(\Q(\sqrt{-11}) \)
Conductor \((165)\)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(170a-547\right){x}-2726a-562\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,1]),K([-547,170]),K([-562,-2726])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,1])),Pol(Vecrev([-547,170])),Pol(Vecrev([-562,-2726]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,1],K![-547,170],K![-562,-2726]]);
 

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: \((-10294453125a+12353343750)\) = \((-a)^{6}\cdot(a-1)^{2}\cdot(-a-1)^{6}\cdot(a-2)^{6}\cdot(-2a+1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 343361479062744140625 \) = \(3^{6}\cdot3^{2}\cdot5^{6}\cdot5^{6}\cdot11^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{32857059929}{125296875} a + \frac{121892766227}{125296875} \)
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 $\left(6 a + 4 : 59 a - 65 : 1\right)$
Height \(0.830616876251219\)
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: $\left(6 a - \frac{109}{4} : -\frac{7}{2} a + \frac{105}{8} : 1\right)$ $\left(-5 a + 3 : 2 a - 2 : 1\right)$
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: \( 0.830616876251219 \)
Period: \( 0.268212646280366 \)
Tamagawa product: \( 192 \)  =  \(2\cdot2\cdot( 2 \cdot 3 )\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.22422170016667 \)
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 \(1\) \(1\) \(2\) \(2\)
\((-a-1)\) \(5\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((a-2)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-2a+1)\) \(11\) \(4\) \(I_2^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)

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
\(3\) 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-d 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 not a \(\Q\)-curve.