Properties

Label 2.2.24.1-48.1-b1
Base field \(\Q(\sqrt{6}) \)
Conductor \((4 a + 12)\)
Conductor norm \( 48 \)
CM no
Base change yes: 48.a6,576.d6
Q-curve yes
Torsion order \( 8 \)
Rank \( 0 \)

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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(Pol(Vecrev([-6, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
 

Weierstrass equation

\(y^2+axy+ay=x^{3}+\left(-a+1\right)x^{2}+\left(-80a+193\right)x-4455a+10911\)
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,1]),K([193,-80]),K([10911,-4455])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([193,-80])),Pol(Vecrev([10911,-4455]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,1],K![193,-80],K![10911,-4455]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4 a + 12)\) = \( \left(-a + 2\right)^{4} \cdot \left(a + 3\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 48 \) = \( 2^{4} \cdot 3 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((209952)\) = \( \left(-a + 2\right)^{10} \cdot \left(a + 3\right)^{16} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 44079842304 \) = \( 2^{10} \cdot 3^{16} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{207646}{6561} \)
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: \(0\)
Torsion structure: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 a - 5 : 29 a - 72 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 5.68350851750159 \)
Tamagawa product: \( 64 \)  =  \(2^{2}\cdot2^{4}\)
Torsion order: \(8\)
Leading coefficient: \(1.16014131805341\)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(2\) \(4\) \(I_{2}^*\) Additive \(-1\) \(4\) \(10\) \(0\)
\( \left(a + 3\right) \) \(3\) \(16\) \(I_{16}\) Split multiplicative \(-1\) \(1\) \(16\) \(16\)

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 48.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 48.a6, 576.d6, defined over \(\Q\), so it is also a \(\Q\)-curve.