Properties

Label 2.0.4.1-67600.6-g5
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 67600 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

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

\({y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}-{x}^{2}+\left(-177i-1716\right){x}-3817i-25925\)
sage: E = EllipticCurve([K([1,1]),K([-1,0]),K([1,1]),K([-1716,-177]),K([-25925,-3817])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([1,1]),Polrev([-1716,-177]),Polrev([-25925,-3817])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,0],K![1,1],K![-1716,-177],K![-25925,-3817]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((100i-240)\) = \((i+1)^{4}\cdot(-i-2)\cdot(2i+1)\cdot(2i+3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 67600 \) = \(2^{4}\cdot5\cdot5\cdot13^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((16742419520i+14955787040)\) = \((i+1)^{10}\cdot(-i-2)^{2}\cdot(2i+1)\cdot(2i+3)^{14}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 503984177369508992000 \) = \(2^{10}\cdot5^{2}\cdot5\cdot13^{14}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{263319363133844}{20393268025} i + \frac{443594369492878}{20393268025} \)
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 $\left(-i - 25 : 25 i + 39 : 1\right)$
Height \(2.7425134625017140073908770415134765567\)
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: $\left(-\frac{5}{2} i - 27 : \frac{57}{4} i + \frac{47}{4} : 1\right)$
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: \( 2.7425134625017140073908770415134765567 \)
Period: \( 0.22171885706780809832484624744295396994 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot1\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 4.8645356031916561022570045224680571136 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(2\) \(I_{2}^{*}\) Additive \(-1\) \(4\) \(10\) \(0\)
\((-i-2)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((2i+1)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((2i+3)\) \(13\) \(4\) \(I_{8}^{*}\) Additive \(1\) \(2\) \(14\) \(8\)

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, 4 and 8.
Its isogeny class 67600.6-g consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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