Properties

Label 2.0.4.1-16562.2-e3
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 16562 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{-1}) \)

Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
 
gp: K = nfinit(Polrev([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}-31293{x}+2081875\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([-31293,0]),K([2081875,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,1]),Polrev([-31293,0]),Polrev([2081875,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![-31293,0],K![2081875,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((91i+91)\) = \((i+1)\cdot(-3i-2)\cdot(2i+3)\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16562 \) = \(2\cdot13\cdot13\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((74853681183008)\) = \((i+1)^{10}\cdot(-3i-2)^{2}\cdot(2i+3)^{2}\cdot(7)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5603073586647405938387928064 \) = \(2^{10}\cdot13^{2}\cdot13^{2}\cdot49^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3389174547561866673}{74853681183008} \)
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 $\left(\frac{2231}{25} : -\frac{1128}{25} i - \frac{9604}{125} : 1\right)$
Height \(1.5893488083448427110307735949810515629\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{55}{2} : -\frac{57}{4} i - \frac{4459}{4} : 1\right)$
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: \( 1.5893488083448427110307735949810515629 \)
Period: \( 0.055022475362562700766137158460427504884 \)
Tamagawa product: \( 480 \)  =  \(( 2 \cdot 5 )\cdot2\cdot2\cdot( 2^{2} \cdot 3 )\)
Torsion order: \(4\)
Leading coefficient: \( 5.2469943389803497515742509144955886548 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)
\((-3i-2)\) \(13\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2i+3)\) \(13\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((7)\) \(49\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)

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 and 4.
Its isogeny class 16562.2-e consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 182.c2
\(\Q\) 1456.i2