Properties

Label 2.0.4.1-57600.2-z1
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 57600 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2={x}^{3}+i{x}^{2}-24{x}-24i\)
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([-24,0]),K([0,-24])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-24,0])),Pol(Vecrev([0,-24]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,1],K![0,0],K![-24,0],K![0,-24]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((240)\) = \((i+1)^{8}\cdot(-i-2)\cdot(2i+1)\cdot(3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 57600 \) = \(2^{8}\cdot5\cdot5\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((960000)\) = \((i+1)^{18}\cdot(-i-2)^{4}\cdot(2i+1)^{4}\cdot(3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 921600000000 \) = \(2^{18}\cdot5^{4}\cdot5^{4}\cdot9\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2863288}{1875} \)
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 $\left(-i - 5 : 5 i - 5 : 1\right)$
Height \(0.443150775611460\)
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: $\left(4 i : 10 i - 10 : 1\right)$
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: \( 0.443150775611460 \)
Period: \( 1.38615777652593 \)
Tamagawa product: \( 64 \)  =  \(2^{2}\cdot2^{2}\cdot2^{2}\cdot1\)
Torsion order: \(4\)
Leading coefficient: \( 4.91421515029857 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(4\) \(I_{6}^{*}\) Additive \(-1\) \(8\) \(18\) \(0\)
\((-i-2)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((2i+1)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((3)\) \(9\) \(1\) \(I_{1}\) 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 57600.2-z consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 960.f4
\(\Q\) 960.o4