Properties

Label 2.2.12.1-192.1-b4
Base field \(\Q(\sqrt{3}) \)
Conductor \((8 a)\)
Conductor norm \( 192 \)
CM no
Base change yes: 96.a2,288.b2
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 3)
 
gp: K = nfinit(a^2 - 3);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\(y^2=x^{3}+\left(a-1\right)x^{2}+\left(-18a-29\right)x+47a+81\)
sage: E = EllipticCurve(K, [0, a - 1, 0, -18*a - 29, 47*a + 81])
 
gp: E = ellinit([0, a - 1, 0, -18*a - 29, 47*a + 81],K)
 
magma: E := ChangeRing(EllipticCurve([0, a - 1, 0, -18*a - 29, 47*a + 81]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((8 a)\) = \( \left(a + 1\right)^{6} \cdot \left(a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 192 \) = \( 2^{6} \cdot 3 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((192)\) = \( \left(a + 1\right)^{12} \cdot \left(a\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 36864 \) = \( 2^{12} \cdot 3^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{140608}{3} \)
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 $\left(5 a + 10 : -22 a - 38 : 1\right)$
Height \(0.620169672849670\)
Torsion structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(1 : -3 a - 5 : 1\right)$ $\left(a + 2 : 0 : 1\right)$
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: \( 0.620169672849670 \)
Period: \( 32.0689894931475 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(8\)
Leading coefficient: \(1.43530826547318\)
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\((j)_{-}\)
\( \left(a + 1\right) \) \(2\) \(4\) \(I_{2}^*\) Additive \(-1\) \(6\) \(12\) \(0\)
\( \left(a\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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

Base change

This curve is the base change of elliptic curves 96.a2, 288.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.