Properties

Label 2.0.3.1-16384.1-e2
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 16384 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

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

\({y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(a+12\right){x}+22a-5\)
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,0]),K([12,1]),K([-5,22])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,1]),Polrev([0,0]),Polrev([12,1]),Polrev([-5,22])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![12,1],K![-5,22]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((128)\) = \((2)^{7}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16384 \) = \(4^{7}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8192)\) = \((2)^{13}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 67108864 \) = \(4^{13}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 23328 \)
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 $\left(-2 a : -2 a + 1 : 1\right)$
Height \(0.87055757458751933367773352763888562286\)
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: $\left(-3 a + 1 : 0 : 1\right)$
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: \( 0.87055757458751933367773352763888562286 \)
Period: \( 2.6383554596354110881005241776926889705 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 2.6521627654373601235836764074864774863 \)
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\((j)_{-}\)
\((2)\) \(4\) \(2\) \(I_{2}^{*}\) 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 16384.1-e consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.