Properties

Label 2.0.4.1-50005.6-a2
Base field \(\Q(\sqrt{-1}) \)
Conductor \((98i-201)\)
Conductor norm \( 50005 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Show commands: Magma / Pari/GP / 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(Pol(Vecrev([1, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2+i{x}{y}+i{y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(-717i-162\right){x}+8651i-2668\)
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([-162,-717]),K([-2668,8651])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([-162,-717])),Pol(Vecrev([-2668,8651]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![-162,-717],K![-2668,8651]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((98i-201)\) = \((2i+1)\cdot(-3i-8)\cdot(4i-11)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 50005 \) = \(5\cdot73\cdot137\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2124538307i+11231775624)\) = \((2i+1)^{20}\cdot(-3i-8)\cdot(4i-11)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 130666446685791015625 \) = \(5^{20}\cdot73\cdot137^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{446734188540318826216893}{130666446685791015625} i - \frac{233523807998465996045776}{130666446685791015625} \)
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: \(0\)
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: $\left(-19 i - \frac{99}{4} : \frac{95}{8} i - \frac{19}{2} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.274908826718102 \)
Tamagawa product: \( 40 \)  =  \(( 2^{2} \cdot 5 )\cdot1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.74908826718102 \)
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\((j)_{-}\)
\((2i+1)\) \(5\) \(20\) \(I_{20}\) Split multiplicative \(-1\) \(1\) \(20\) \(20\)
\((-3i-8)\) \(73\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((4i-11)\) \(137\) \(2\) \(I_{2}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 50005.6-a consists of curves linked by isogenies of degree 2.

Base change

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