Properties

Label 2.0.4.1-82369.3-b1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 82369 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+i{x}{y}={x}^{3}+{x}^{2}+\left(57i-8\right){x}+1917i-3083\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([-8,57]),K([-3083,1917])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([-8,57]),Polrev([-3083,1917])], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![-8,57],K![-3083,1917]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((63i-280)\) = \((4i+5)^{2}\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 82369 \) = \(41^{2}\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5076157401i-2541737800)\) = \((4i+5)^{10}\cdot(7)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 32227805003675914801 \) = \(41^{10}\cdot49^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{18657297}{138462289} i + \frac{73533528}{19780327} \)
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 $\left(55 i - \frac{27}{2} : -\frac{693}{4} i + \frac{1629}{4} : 1\right)$
Height \(1.1816083713606643393842235771943204039\)
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: $\left(-4 i + \frac{61}{4} : -\frac{61}{8} i - 2 : 1\right)$
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: \( 1.1816083713606643393842235771943204039 \)
Period: \( 0.33233008323932537877429381547232001080 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.5707360336422930955258664476347112513 \)
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\((j)_{-}\)
\((4i+5)\) \(41\) \(4\) \(I_{4}^{*}\) Additive \(1\) \(2\) \(10\) \(4\)
\((7)\) \(49\) \(2\) \(I_{2}\) Non-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 82369.3-b consists of curves linked by isogenies of degree 2.

Base change

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

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