Properties

Label 2.0.8.1-8712.5-f1
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 8712 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2+a{x}{y}={x}^{3}-1994{x}+35226\)
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,0]),K([-1994,0]),K([35226,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-1994,0]),Polrev([35226,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,0],K![0,0],K![-1994,0],K![35226,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((66a)\) = \((a)^{3}\cdot(-a-1)\cdot(a-1)\cdot(a+3)\cdot(a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 8712 \) = \(2^{3}\cdot3\cdot3\cdot11\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-18519655968)\) = \((a)^{10}\cdot(-a-1)^{14}\cdot(a-1)^{14}\cdot(a+3)^{2}\cdot(a-3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 342977657173078017024 \) = \(2^{10}\cdot3^{14}\cdot3^{14}\cdot11^{2}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{27403349188178}{578739249} \)
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 $\left(-2 : a + 198 : 1\right)$
Height \(0.17823931261394709869304471356601762574\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{103}{2} : \frac{103}{4} a : 1\right)$ $\left(-2 a + 26 : -13 a - 2 : 1\right)$
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: \( 0.17823931261394709869304471356601762574 \)
Period: \( 0.21840046480268000328359450450026687437 \)
Tamagawa product: \( 1568 \)  =  \(2\cdot( 2 \cdot 7 )\cdot( 2 \cdot 7 )\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 5.3950830004148854087136537038153876213 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(III^{*}\) Additive \(1\) \(3\) \(10\) \(0\)
\((-a-1)\) \(3\) \(14\) \(I_{14}\) Split multiplicative \(-1\) \(1\) \(14\) \(14\)
\((a-1)\) \(3\) \(14\) \(I_{14}\) Split multiplicative \(-1\) \(1\) \(14\) \(14\)
\((a+3)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-3)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 8712.5-f consists of curves linked by isogenies of degrees dividing 4.

Base change

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

Base field Curve
\(\Q\) 264.d2
\(\Q\) 2112.p2