Properties

Label 2.2.12.1-768.1-e6
Base field \(\Q(\sqrt{3}) \)
Conductor norm \( 768 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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Show commands: Magma / PariGP / SageMath

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(Polrev([-3, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(258a-449\right){x}+3043a-5271\)
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-449,258]),K([-5271,3043])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-449,258]),Polrev([-5271,3043])], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-449,258],K![-5271,3043]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((16a)\) = \((a+1)^{8}\cdot(a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 768 \) = \(2^{8}\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3072)\) = \((a+1)^{20}\cdot(a)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 9437184 \) = \(2^{20}\cdot3^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{28756228}{3} \)
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: \(0\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(5 a - 9 : 0 : 1\right)$ $\left(-10 a + 17 : 0 : 1\right)$
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: \( 2.8417542587507934505185738139717168448 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.6406875862605360264830853899829179811 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((a+1)\) \(2\) \(4\) \(I_{8}^{*}\) Additive \(1\) \(8\) \(20\) \(0\)
\((a)\) \(3\) \(2\) \(I_{2}\) 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, 4 and 8.
Its isogeny class 768.1-e consists of curves linked by isogenies of degrees dividing 16.

Base change

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

Base field Curve
\(\Q\) 192.b2
\(\Q\) 576.d2