Properties

Label 2.0.4.1-92416.1-d2
Base field \(\Q(\sqrt{-1}) \)
Conductor \((304)\)
Conductor norm \( 92416 \)
CM no
Base change yes: 1216.e1,1216.m1
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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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

\({y}^2={x}^{3}-i{x}^{2}+1368{x}-157168i\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([1368,0]),K([0,-157168])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([1368,0])),Pol(Vecrev([0,-157168]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![1368,0],K![0,-157168]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((304)\) = \((i+1)^{8}\cdot(19)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 92416 \) = \(2^{8}\cdot361\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((10445360463872)\) = \((i+1)^{78}\cdot(19)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 109105555220220283017232384 \) = \(2^{78}\cdot361\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{69173457625}{2550136832} \)
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: \(2\)
Generators $\left(\frac{2060}{9} i : \frac{65536}{27} i - \frac{65536}{27} : 1\right)$ $\left(-\frac{3348}{49} i : \frac{65536}{343} i + \frac{65536}{343} : 1\right)$
Heights \(1.69840506937468\) \(2.50432828474085\)
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: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 4.25336385418227 \)
Period: \( 0.0947386166594565 \)
Tamagawa product: \( 4 \)  =  \(2^{2}\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 6.44732492311299 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(4\) \(I_{66}^{*}\) Additive \(1\) \(8\) \(78\) \(54\)
\((19)\) \(361\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 92416.1-d consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of 1216.e1, 1216.m1, defined over \(\Q\), so it is also a \(\Q\)-curve.