Properties

Label 2.0.4.1-57800.6-c1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 57800 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((70i+230)\) = \((i+1)^{3}\cdot(-i-2)\cdot(2i+1)\cdot(i-4)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 57800 \) = \(2^{3}\cdot5\cdot5\cdot17^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-160i+300)\) = \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(i-4)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 115600 \) = \(2^{4}\cdot5\cdot5\cdot17^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 2048 i - \frac{6144}{5} \)
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 $\left(1 : 0 : 1\right)$
Height \(0.20745637874011153748613369846326547130\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.20745637874011153748613369846326547130 \)
Period: \( 5.0296459569020130581948882843303652750 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 4.1737285462549389285419191018643459415 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(2\) \(III\) Additive \(-1\) \(3\) \(4\) \(0\)
\((-i-2)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((2i+1)\) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((i-4)\) \(17\) \(1\) \(II\) Additive \(1\) \(2\) \(2\) \(0\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 57800.6-c consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

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