Properties

Label 2.0.11.1-27225.5-n4
Base field \(\Q(\sqrt{-11}) \)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

\({y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-8550a+7739\right){x}-169106a+696439\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([1,1]),K([7739,-8550]),K([696439,-169106])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,-1]),Polrev([1,1]),Polrev([7739,-8550]),Polrev([696439,-169106])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![1,1],K![7739,-8550],K![696439,-169106]]);
 

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: \((-269624410110a-42152244255)\) = \((-a)^{2}\cdot(a-1)^{20}\cdot(-a-1)\cdot(a-2)^{4}\cdot(-2a+1)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 231234053269296602559375 \) = \(3^{2}\cdot3^{20}\cdot5\cdot5^{4}\cdot11^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{101079974296908724042}{263688070325625} a + \frac{11894483254921076399}{87896023441875} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{133}{4} a - \frac{113}{4} : \frac{121}{4} a - \frac{403}{8} : 1\right)$
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: \( 0.10119875753379498072382211507809859372 \)
Tamagawa product: \( 320 \)  =  \(2\cdot( 2^{2} \cdot 5 )\cdot1\cdot2\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 4.8820117525781777309720722780435083797 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a-1)\) \(3\) \(20\) \(I_{20}\) Split multiplicative \(-1\) \(1\) \(20\) \(20\)
\((-a-1)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((a-2)\) \(5\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-2a+1)\) \(11\) \(4\) \(I_{3}^{*}\) Additive \(-1\) \(2\) \(9\) \(3\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 27225.5-n consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.