Properties

Label 2.0.4.1-67600.6-i8
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 67600 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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

\({y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-36331i-787\right){x}+1957246i-1804685\)
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([0,0]),K([-787,-36331]),K([-1804685,1957246])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([0,0]),Polrev([-787,-36331]),Polrev([-1804685,1957246])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,1],K![0,0],K![-787,-36331],K![-1804685,1957246]]);
 

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: \((-231826000i+397928000)\) = \((i+1)^{8}\cdot(-i-2)^{4}\cdot(2i+1)^{3}\cdot(2i+3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 212089987460000000 \) = \(2^{8}\cdot5^{4}\cdot5^{3}\cdot13^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{14159685840327748}{1373125} i + \frac{7060801251114256}{1373125} \)
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 $\left(\frac{11431}{150} i + \frac{71123}{900} : -\frac{276458}{3375} i + \frac{8741}{9000} : 1\right)$
Height \(6.9993748452367707654490313501593112517\)
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: $\left(80 i + 81 : -154 i - 10 : 1\right)$
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: \( 6.9993748452367707654490313501593112517 \)
Period: \( 0.11021910992577144587358562037137029732 \)
Tamagawa product: \( 48 \)  =  \(1\cdot2^{2}\cdot3\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 4.6287891928729868309465404522743628462 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(1\) \(I_0^{*}\) Additive \(1\) \(4\) \(8\) \(0\)
\((-i-2)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((2i+1)\) \(5\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((2i+3)\) \(13\) \(4\) \(I_{3}^{*}\) Additive \(1\) \(2\) \(9\) \(3\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 67600.6-i consists of curves linked by isogenies of degrees dividing 12.

Base change

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

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