Properties

Label 2.0.8.1-36864.2-m2
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 36864 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{-2}) \)

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

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((192)\) = \((a)^{12}\cdot(-a-1)\cdot(a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 36864 \) = \(2^{12}\cdot3\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1368576a+124416)\) = \((a)^{18}\cdot(-a-1)^{5}\cdot(a-1)^{10}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3761479876608 \) = \(2^{18}\cdot3^{5}\cdot3^{10}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{27353216}{59049} a + \frac{95630912}{59049} \)
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 $\left(a - 3 : 6 a - 6 : 1\right)$ $\left(7 a + 9 : 36 a + 18 : 1\right)$
Heights \(0.19202783583907219209968462135853351167\) \(0.31562074836709545701878007553678676627\)
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: $\left(1 : 0 : 1\right)$
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: \( 0.056789167248051816526922264966444345063 \)
Period: \( 1.2232687065797177506282314228663949634 \)
Tamagawa product: \( 200 \)  =  \(2^{2}\cdot5\cdot( 2 \cdot 5 )\)
Torsion order: \(2\)
Leading coefficient: \( 9.8243169229254774904948841676541720977 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_{2}^{*}\) Additive \(-1\) \(12\) \(18\) \(0\)
\((-a-1)\) \(3\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((a-1)\) \(3\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)

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 36864.2-m 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.