Properties

Label 2.0.3.1-49152.1-b1
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 49152 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

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Show commands: Magma / PariGP / SageMath

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

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

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

Weierstrass equation

\({y}^2={x}^{3}+\left(-a+1\right){x}^{2}-7a{x}+9\)
sage: E = EllipticCurve([K([0,0]),K([1,-1]),K([0,0]),K([0,-7]),K([9,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,-1]),Polrev([0,0]),Polrev([0,-7]),Polrev([9,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,-1],K![0,0],K![0,-7],K![9,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-256a+128)\) = \((-2a+1)\cdot(2)^{7}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49152 \) = \(3\cdot4^{7}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-73728)\) = \((-2a+1)^{4}\cdot(2)^{13}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5435817984 \) = \(3^{4}\cdot4^{13}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{4000}{9} \)
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: \(2\)
Generators $\left(5 a - 5 : 12 : 1\right)$ $\left(-3 a + 3 : 8 a - 4 : 1\right)$
Heights \(0.31973282060884456447180241516890429164\) \(0.60376261687600887962194780136357871108\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 1 : 0 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.19304272447194349694279523362712900262 \)
Period: \( 2.1462940348877933338525230870475366722 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 3.8273837806794838567859866031835727132 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((2)\) \(4\) \(4\) \(I_{2}^{*}\) Additive \(-1\) \(7\) \(13\) \(0\)

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.
Its isogeny class 49152.1-b consists of curves linked by isogenies of degree 2.

Base change

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

Base field Curve
\(\Q\) 384.b2
\(\Q\) 1152.j2