Properties

Label 2.0.11.1-27225.1-c3
Base field \(\Q(\sqrt{-11}) \)
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(Polrev([3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 

Weierstrass equation

\({y}^2+{y}={x}^{3}-a{x}^{2}+\left(-29a-12\right){x}+238a-163\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([1,0]),K([-12,-29]),K([-163,238])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([1,0]),Polrev([-12,-29]),Polrev([-163,238])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![1,0],K![-12,-29],K![-163,238]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-88a-33)\) = \((-a)^{2}\cdot(-a-1)^{2}\cdot(-2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27225 \) = \(3^{2}\cdot5^{2}\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3396712a+47392917)\) = \((-a)^{6}\cdot(-a-1)^{6}\cdot(-2a+1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2441681628890625 \) = \(3^{6}\cdot5^{6}\cdot11^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{4096}{11} \)
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 $\left(-2 a - 8 : 11 a - 6 : 1\right)$
Height \(0.94896436445216592754320738556785324031\)
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: \( 0.94896436445216592754320738556785324031 \)
Period: \( 0.72071237796473239309207557488464487097 \)
Tamagawa product: \( 4 \)  =  \(1\cdot1\cdot2^{2}\)
Torsion order: \(1\)
Leading coefficient: \( 3.2994042169470655649804915320258695414 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(1\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((-a-1)\) \(5\) \(1\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((-2a+1)\) \(11\) \(4\) \(I_{2}^{*}\) 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
\(2\) 2Cn
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5 and 25.
Its isogeny class 27225.1-c consists of curves linked by isogenies of degrees dividing 25.

Base change

This elliptic curve is a \(\Q\)-curve.

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