Properties

Label 2.0.3.1-57600.1-k4
Base field \(\Q(\sqrt{-3}) \)
Conductor \((240)\)
Conductor norm \( 57600 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(482a-481\right){x}+2329a-1405\)
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,0]),K([-481,482]),K([-1405,2329])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-481,482])),Pol(Vecrev([-1405,2329]))], K);
 
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![-481,482],K![-1405,2329]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((240)\) = \((-2a+1)^{2}\cdot(2)^{4}\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 57600 \) = \(3^{2}\cdot4^{4}\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-5598720000)\) = \((-2a+1)^{14}\cdot(2)^{12}\cdot(5)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 31345665638400000000 \) = \(3^{14}\cdot4^{12}\cdot25^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{111284641}{50625} \)
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 $\left(18 a - 15 : -74 a + 90 : 1\right)$
Height \(2.66466074092007\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(11 a - 23 : 0 : 1\right)$ $\left(2 a - 5 : 0 : 1\right)$
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: \( 2.66466074092007 \)
Period: \( 0.322695746401859 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 3.97159105467979 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(4\) \(I_8^{*}\) Additive \(-1\) \(2\) \(14\) \(8\)
\((2)\) \(4\) \(4\) \(I_4^{*}\) Additive \(-1\) \(4\) \(12\) \(0\)
\((5)\) \(25\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 57600.1-k consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.