Properties

Label 2.0.11.1-27225.6-e1
Base field \(\Q(\sqrt{-11}) \)
Conductor \((-99a+33)\)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
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

\({y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-10a+3\right){x}-8a+8\)
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([1,0]),K([3,-10]),K([8,-8])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,1])),Pol(Vecrev([1,0])),Pol(Vecrev([3,-10])),Pol(Vecrev([8,-8]))], K);
 
magma: E := EllipticCurve([K![0,0],K![1,1],K![1,0],K![3,-10],K![8,-8]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-99a+33)\) = \((-a)\cdot(a-1)\cdot(a-2)^{2}\cdot(-2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27225 \) = \(3\cdot3\cdot5^{2}\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-235224a-3267)\) = \((-a)^{3}\cdot(a-1)^{3}\cdot(a-2)^{6}\cdot(-2a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 166770140625 \) = \(3^{3}\cdot3^{3}\cdot5^{6}\cdot11^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{45056}{27} \)
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 $\left(-6 : 8 a + 1 : 1\right)$
Height \(0.107600940988944\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.107600940988944 \)
Period: \( 1.59696018929354 \)
Tamagawa product: \( 18 \)  =  \(3\cdot1\cdot2\cdot3\)
Torsion order: \(1\)
Leading coefficient: \( 3.73032192560579 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a-1)\) \(3\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((a-2)\) \(5\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((-2a+1)\) \(11\) \(3\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 27225.6-e consists of this curve only.

Base change

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