Properties

Label 2.0.4.1-57600.2-a2
Base field \(\Q(\sqrt{-1}) \)
Conductor norm \( 57600 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

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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(Polrev([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}-i{x}^{2}-80{x}+80i\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([-80,0]),K([0,80])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([-80,0]),Polrev([0,80])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![-80,0],K![0,80]]);
 

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: \((33592320)\) = \((i+1)^{20}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1128443962982400 \) = \(2^{20}\cdot5\cdot5\cdot9^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{54607676}{32805} \)
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 $\left(4 i : 12 i - 12 : 1\right)$ $\left(i + 9 : 9 i + 9 : 1\right)$
Heights \(0.38023225445235132440091342653239198510\) \(0.57311442611204913160991352285305197984\)
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: $\left(i : 0 : 1\right)$
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: \( 0.21791659029974996755583493294401810290 \)
Period: \( 0.76578751017881994108376081967683610714 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot1\cdot1\cdot2^{3}\)
Torsion order: \(2\)
Leading coefficient: \( 5.3400896995937124197036464962557196405 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(4\) \(I_{8}^{*}\) Additive \(1\) \(8\) \(20\) \(0\)
\((-i-2)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((2i+1)\) \(5\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((3)\) \(9\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)

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, 4 and 8.
Its isogeny class 57600.2-a consists of curves linked by isogenies of degrees dividing 8.

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.c6
\(\Q\) 960.j6