Properties

Label 2.0.4.1-97344.2-b1
Base field \(\Q(\sqrt{-1}) \)
Conductor \((312)\)
Conductor norm \( 97344 \)
CM no
Base change yes: 1248.b4,1248.f4
Q-curve yes
Torsion order \( 4 \)
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

\({y}^2+\left(i+1\right){x}{y}={x}^{3}-i{x}^{2}+26{x}+357i\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([0,0]),K([26,0]),K([0,357])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([26,0])),Pol(Vecrev([0,357]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![0,0],K![26,0],K![0,357]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((312)\) = \((i+1)^{6}\cdot(3)\cdot(-3i-2)\cdot(2i+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 97344 \) = \(2^{6}\cdot9\cdot13\cdot13\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((55269864)\) = \((i+1)^{6}\cdot(3)^{12}\cdot(-3i-2)\cdot(2i+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3054757866578496 \) = \(2^{6}\cdot9^{12}\cdot13\cdot13\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{245314376}{6908733} \)
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 $\left(-\frac{223}{50} i + \frac{243}{25} : \frac{7109}{500} i - \frac{17963}{500} : 1\right)$ $\left(-\frac{13}{8} i : -\frac{379}{32} i - \frac{431}{32} : 1\right)$
Heights \(2.13836744380331\) \(1.56047198015838\)
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: $\left(-5 i : -11 i - 16 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 2.72809427912311 \)
Period: \( 0.717282506493570 \)
Tamagawa product: \( 12 \)  =  \(1\cdot( 2^{2} \cdot 3 )\cdot1\cdot1\)
Torsion order: \(4\)
Leading coefficient: \( 5.87044290744058 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(1\) \(II\) Additive \(1\) \(6\) \(6\) \(0\)
\((3)\) \(9\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((-3i-2)\) \(13\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((2i+3)\) \(13\) \(1\) \(I_{1}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 97344.2-b consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of 1248.b4, 1248.f4, defined over \(\Q\), so it is also a \(\Q\)-curve.