Properties

Label 2.2.12.1-3600.1-j4
Base field \(\Q(\sqrt{3}) \)
Conductor \((60)\)
Conductor norm \( 3600 \)
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 \( x^{2} - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-3, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-122\right){x}-116a-1\)
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([1,1]),K([-122,-1]),K([-1,-116])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([1,1])),Pol(Vecrev([-122,-1])),Pol(Vecrev([-1,-116]))], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![1,1],K![-122,-1],K![-1,-116]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((60)\) = \((a+1)^{4}\cdot(a)^{2}\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 3600 \) = \(2^{4}\cdot3^{2}\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((87480000)\) = \((a+1)^{12}\cdot(a)^{14}\cdot(5)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 7652750400000000 \) = \(2^{12}\cdot3^{14}\cdot25^{4}\)
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 $\left(8 a + 27 : -99 a - 134 : 1\right)$
Height \(1.33233037046003\)
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: $\left(-\frac{11}{2} a : \frac{9}{4} a + \frac{31}{4} : 1\right)$ $\left(-a : 1 : 1\right)$
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: \( 1.33233037046003 \)
Period: \( 2.58156597121488 \)
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\((j)_{-}\)
\((a+1)\) \(2\) \(4\) \(I_{4}^{*}\) Additive \(1\) \(4\) \(12\) \(0\)
\((a)\) \(3\) \(4\) \(I_{8}^{*}\) Additive \(-1\) \(2\) \(14\) \(8\)
\((5)\) \(25\) \(2\) \(I_{4}\) 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 3600.1-j 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.