Properties

Label 2.0.8.1-13122.5-d4
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 13122 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 3 \)
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

\({y}^2+{x}{y}={x}^{3}-{x}^{2}+3{x}-1\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([3,0]),K([-1,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,0]),Polrev([3,0]),Polrev([-1,0])], K);
 
magma: E := EllipticCurve([K![1,0],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: \((81a)\) = \((a)\cdot(-a-1)^{4}\cdot(a-1)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 13122 \) = \(2\cdot3^{4}\cdot3^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1458)\) = \((a)^{2}\cdot(-a-1)^{6}\cdot(a-1)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2125764 \) = \(2^{2}\cdot3^{6}\cdot3^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3375}{2} \)
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 $\left(a : -a - 1 : 1\right)$
Height \(0.30015996156315906378013457433845531587\)
Torsion structure: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(1 : -2 : 1\right)$
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: \( 0.30015996156315906378013457433845531587 \)
Period: \( 4.0843965388841212595817515611911186142 \)
Tamagawa product: \( 18 \)  =  \(2\cdot3\cdot3\)
Torsion order: \(3\)
Leading coefficient: \( 3.4675733304747480324358693131942067062 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-a-1)\) \(3\) \(3\) \(IV\) Additive \(-1\) \(4\) \(6\) \(0\)
\((a-1)\) \(3\) \(3\) \(IV\) Additive \(-1\) \(4\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cn
\(3\) 3B.1.1
\(7\) 7B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 7 and 21.
Its isogeny class 13122.5-d consists of curves linked by isogenies of degrees dividing 21.

Base change

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

Base field Curve
\(\Q\) 162.b4
\(\Q\) 5184.p4