Properties

Label 2.0.11.1-27225.9-e1
Base field \(\Q(\sqrt{-11}) \)
Conductor \((88a-121)\)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve no
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

\({y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(1371a-855\right){x}+19460a+15212\)
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([0,1]),K([-855,1371]),K([15212,19460])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([-855,1371])),Pol(Vecrev([15212,19460]))], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,1],K![0,1],K![-855,1371],K![15212,19460]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((88a-121)\) = \((a-1)^{2}\cdot(a-2)^{2}\cdot(-2a+1)^{2}\)
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: \((-66660473a+162895766)\) = \((a-1)^{9}\cdot(a-2)^{4}\cdot(-2a+1)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 29007177751220625 \) = \(3^{9}\cdot5^{4}\cdot11^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{336572416}{121} a - \frac{914255872}{121} \)
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 $\left(\frac{69}{4} a - \frac{175}{4} : -\frac{255}{4} a - \frac{957}{8} : 1\right)$
Height \(2.76211947320061\)
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: \( 2.76211947320061 \)
Period: \( 0.301139804549088 \)
Tamagawa product: \( 8 \)  =  \(2\cdot1\cdot2^{2}\)
Torsion order: \(1\)
Leading coefficient: \( 8.02535513303434 \)
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\((j)_{-}\)
\((a-1)\) \(3\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)
\((a-2)\) \(5\) \(1\) \(IV\) Additive \(-1\) \(2\) \(4\) \(0\)
\((-2a+1)\) \(11\) \(4\) \(I_3^{*}\) 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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 27225.9-e consists of curves linked by isogenies of degree 3.

Base change

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