Properties

Label 2.0.11.1-27225.4-f7
Base field \(\Q(\sqrt{-11}) \)
Conductor norm \( 27225 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
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

\({y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(2756a-1351\right){x}-49535a-62584\)
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([0,0]),K([-1351,2756]),K([-62584,-49535])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([0,0]),Polrev([-1351,2756]),Polrev([-62584,-49535])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,1],K![0,0],K![-1351,2756],K![-62584,-49535]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((99a-66)\) = \((-a)\cdot(a-1)\cdot(-a-1)^{2}\cdot(-2a+1)^{2}\)
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: \((7870203a+15416973)\) = \((-a)^{4}\cdot(a-1)^{5}\cdot(-a-1)^{6}\cdot(-2a+1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 544838049421875 \) = \(3^{4}\cdot3^{5}\cdot5^{6}\cdot11^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{54238838797}{243} a + \frac{1331574640}{9} \)
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 $\left(11 a - 10 : 252 a - 250 : 1\right)$
Height \(2.8442268042642406084306206819437615363\)
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: $\left(-17 a + \frac{143}{4} : \frac{17}{2} a - \frac{143}{8} : 1\right)$
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: \( 2.8442268042642406084306206819437615363 \)
Period: \( 0.26737169462912409322884330718055061321 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot1\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 3.6686247670926623025179502049052719676 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a-1)\) \(3\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((-a-1)\) \(5\) \(2\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)
\((-2a+1)\) \(11\) \(2\) \(I_0^{*}\) 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
\(2\) 2B
\(5\) 5B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 5, 10 and 20.
Its isogeny class 27225.4-f consists of curves linked by isogenies of degrees dividing 20.

Base change

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

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