Properties

Label 2.0.11.1-27225.2-e1
Base field \(\Q(\sqrt{-11}) \)
Conductor \((55a-165)\)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-408a+493\right){x}-784a+7502\)
sage: E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([493,-408]),K([7502,-784])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([493,-408])),Pol(Vecrev([7502,-784]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![493,-408],K![7502,-784]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((55a-165)\) = \((-a)^{2}\cdot(-a-1)\cdot(a-2)\cdot(-2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27225 \) = \(3^{2}\cdot5\cdot5\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((205632845a-192602355)\) = \((-a)^{3}\cdot(-a-1)^{8}\cdot(a-2)\cdot(-2a+1)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 124344897767578125 \) = \(3^{3}\cdot5^{8}\cdot5\cdot11^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{24171483673}{390625} a - \frac{16292777334}{390625} \)
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: $\left(-7 a - \frac{21}{4} : \frac{7}{2} a + \frac{17}{8} : 1\right)$
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: \( 0.392054257420513 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.89134090083673 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((-a)\) \(3\) \(2\) \(III\) Additive \(1\) \(2\) \(3\) \(0\)
\((-a-1)\) \(5\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((a-2)\) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((-2a+1)\) \(11\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 27225.2-e consists of curves linked by isogenies of degree 2.

Base change

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