Properties

Label 2.2.24.1-726.1-d4
Base field \(\Q(\sqrt{6}) \)
Conductor norm \( 726 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 6 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{6}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 6 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
 
gp: K = nfinit(Polrev([-6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(1611a-3944\right){x}-54572a+133674\)
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([-3944,1611]),K([133674,-54572])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-3944,1611]),Polrev([133674,-54572])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![-3944,1611],K![133674,-54572]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-11a)\) = \((-a+2)\cdot(a+3)\cdot(11)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 726 \) = \(2\cdot3\cdot121\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((255552)\) = \((-a+2)^{12}\cdot(a+3)^{2}\cdot(11)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 65306824704 \) = \(2^{12}\cdot3^{2}\cdot121^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{57736239625}{255552} \)
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 $\left(8 a - 20 : -141 a + 346 : 1\right)$
Height \(0.77416442216064295037161700101522287494\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 a - 6 : 123 a - 300 : 1\right)$
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: \( 0.77416442216064295037161700101522287494 \)
Period: \( 9.7783813800140005545659847615063526149 \)
Tamagawa product: \( 72 \)  =  \(( 2^{2} \cdot 3 )\cdot2\cdot3\)
Torsion order: \(6\)
Leading coefficient: \( 6.1809403309634603721807794037038304370 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((a+3)\) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((11)\) \(121\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 726.1-d consists of curves linked by isogenies of degrees dividing 6.

Base change

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

Base field Curve
\(\Q\) 198.e1
\(\Q\) 2112.v1