Properties

Label 2.0.11.1-27225.6-c1
Base field \(\Q(\sqrt{-11}) \)
Conductor \((-99a+33)\)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
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

\({y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-171a+85\right){x}+750a-1899\)
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([0,0]),K([85,-171]),K([-1899,750])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([85,-171])),Pol(Vecrev([-1899,750]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,-1],K![0,0],K![85,-171],K![-1899,750]]);
 

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: \((64243377a-90094059)\) = \((-a)^{10}\cdot(a-1)^{2}\cdot(a-2)^{6}\cdot(-2a+1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 14710627334390625 \) = \(3^{10}\cdot3^{2}\cdot5^{6}\cdot11^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{349209575}{59049} a - \frac{1245002978}{59049} \)
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 $\left(3 a - 32 : -106 a + 5 : 1\right)$
Height \(1.42211340213212\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-8 a - 10 : 4 a + 5 : 1\right)$ $\left(5 a + \frac{11}{4} : -\frac{5}{2} a - \frac{11}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.42211340213212 \)
Period: \( 0.534743389258248 \)
Tamagawa product: \( 64 \)  =  \(2\cdot2\cdot2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 3.66862476709266 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(2\) \(I_{10}\) Non-split multiplicative \(1\) \(1\) \(10\) \(10\)
\((a-1)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((a-2)\) \(5\) \(4\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((-2a+1)\) \(11\) \(4\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(5\) 5B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 5 and 10.
Its isogeny class 27225.6-c consists of curves linked by isogenies of degrees dividing 20.

Base change

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