Properties

Label 2.0.4.1-88200.2-k3
Base field \(\Q(\sqrt{-1}) \)
Conductor \((210 i + 210)\)
Conductor norm \( 88200 \)
CM no
Base change yes: 840.f2,1680.p2
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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

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

sage: x = polygen(QQ); K.<i> = NumberField(x^2 + 1)
 
gp: K = nfinit(i^2 + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<i> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\(y^2+\left(i+1\right)xy+\left(i+1\right)y=x^{3}+\left(-i+430\right)x-2013i\)
sage: E = EllipticCurve(K, [i + 1, 0, i + 1, -i + 430, -2013*i])
 
gp: E = ellinit([i + 1, 0, i + 1, -i + 430, -2013*i],K)
 
magma: E := ChangeRing(EllipticCurve([i + 1, 0, i + 1, -i + 430, -2013*i]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((210 i + 210)\) = \( \left(i + 1\right)^{3} \cdot \left(3\right) \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(7\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 88200 \) = \( 2^{3} \cdot 5^{2} \cdot 9 \cdot 49 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((3214890000)\) = \( \left(i + 1\right)^{8} \cdot \left(3\right)^{8} \cdot \left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(7\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 10335517712100000000 \) = \( 2^{8} \cdot 5^{8} \cdot 9^{8} \cdot 49^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{549871953124}{200930625} \)
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 $\left(355 i : -4903 i + 4902 : 1\right)$
Height \(0.902352652534272\)
Torsion structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-5 i : 47 i - 48 : 1\right)$ $\left(5 i : -3 i + 2 : 1\right)$
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: \( 0.902352652534272 \)
Period: \( 0.350216404766139 \)
Tamagawa product: \( 512 \)  =  \(2\cdot2^{2}\cdot2^{2}\cdot2^{3}\cdot2\)
Torsion order: \(8\)
Leading coefficient: \(5.05629922882787\)
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\((j)_{-}\)
\( \left(i + 1\right) \) \(2\) \(2\) \(I_{1}^*\) Additive \(1\) \(3\) \(8\) \(0\)
\( \left(-i - 2\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(2 i + 1\right) \) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(3\right) \) \(9\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\( \left(7\right) \) \(49\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 88200.2-k consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 840.f2, 1680.p2, defined over \(\Q\), so it is also a \(\Q\)-curve.