Properties

Label 2.2.8.1-196.3-a4
Base field \(\Q(\sqrt{2}) \)
Conductor norm \( 196 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+a{x}{y}={x}^{3}-a{x}^{2}+\left(-6a-110\right){x}+2170a-2612\)
sage: E = EllipticCurve([K([0,1]),K([0,-1]),K([0,0]),K([-110,-6]),K([-2612,2170])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([0,-1]),Polrev([0,0]),Polrev([-110,-6]),Polrev([-2612,2170])], K);
 
magma: E := EllipticCurve([K![0,1],K![0,-1],K![0,0],K![-110,-6],K![-2612,2170]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((10a-2)\) = \((a)^{2}\cdot(2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 196 \) = \(2^{2}\cdot7^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-187532096a+698004624)\) = \((a)^{8}\cdot(2a+1)^{18}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -416873881065074944 \) = \(-2^{8}\cdot7^{18}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{29518306565684}{13841287201} a + \frac{41622722395132}{13841287201} \)
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: \(0\)
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: $\left(-11 a + \frac{7}{2} : -\frac{7}{4} a + 11 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 1.1104405915832555608038747079719409581 \)
Tamagawa product: \( 12 \)  =  \(3\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 1.1778001086199822146816645485956046400 \)
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\((j)_{-}\)
\((a)\) \(2\) \(3\) \(IV^{*}\) Additive \(-1\) \(2\) \(8\) \(0\)
\((2a+1)\) \(7\) \(4\) \(I_{12}^{*}\) Additive \(-1\) \(2\) \(18\) \(12\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 196.3-a consists of curves linked by isogenies of degrees dividing 12.

Base change

This elliptic curve is not a \(\Q\)-curve.

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