Properties

Label 2.0.8.1-41616.5-b3
Base field \(\Q(\sqrt{-2}) \)
Conductor \((204)\)
Conductor norm \( 41616 \)
CM no
Base change yes: 3264.c1,816.f1
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

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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

\({y}^2+a{x}{y}={x}^{3}+{x}^{2}-188{x}-1020\)
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([-188,0]),K([-1020,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-188,0])),Pol(Vecrev([-1020,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![-188,0],K![-1020,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((204)\) = \((a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41616 \) = \(2^{4}\cdot3\cdot3\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3569184)\) = \((a)^{10}\cdot(-a-1)^{8}\cdot(a-1)^{8}\cdot(-2a+3)\cdot(2a+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 12739074425856 \) = \(2^{10}\cdot3^{8}\cdot3^{8}\cdot17\cdot17\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{22994537186}{111537} \)
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 $\left(-2 a - 12 : 20 a - 18 : 1\right)$ $\left(-8 : 4 a - 2 : 1\right)$
Heights \(0.721288796722825\) \(0.973742777986768\)
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: $\left(-\frac{17}{2} : \frac{17}{4} a : 1\right)$
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: \( 0.612725906293986 \)
Period: \( 0.809400023130075 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2\cdot2\cdot1\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 5.61092469677120 \)
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 \(-1\) \(4\) \(10\) \(0\)
\((-a-1)\) \(3\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((a-1)\) \(3\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((-2a+3)\) \(17\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((2a+3)\) \(17\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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, 4 and 8.
Its isogeny class 41616.5-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of 3264.c1, 816.f1, defined over \(\Q\), so it is also a \(\Q\)-curve.