Properties

Label 2.2.8.1-1024.1-d4
Base field \(\Q(\sqrt{2}) \)
Conductor \((32)\)
Conductor norm \( 1024 \)
CM yes (\(-8\))
Base change yes: 256.d1,256.a2
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / 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(Pol(Vecrev([-2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((32)\) = \((a)^{10}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1024 \) = \(2^{10}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((512)\) = \((a)^{18}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 262144 \) = \(2^{18}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 8000 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-2}]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-1 : -2 : 1\right)$
Height \(0.480125797508298\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-a - 1 : 0 : 1\right)$ $\left(a - 1 : 0 : 1\right)$
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: \( 0.480125797508298 \)
Period: \( 25.3799738685968 \)
Tamagawa product: \( 4 \)
Torsion order: \(4\)
Leading coefficient: \( 2.15412659703815 \)
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\((j)_{-}\)
\((a)\) \(2\) \(4\) \(I_4^{*}\) Additive \(-1\) \(10\) \(18\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 1024.1-d consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of 256.d1, 256.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.