Properties

Label 2.0.8.1-5202.5-d5
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 5202 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 12 \)
Rank \( 1 \)

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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

\({y}^2+{x}{y}+{y}={x}^{3}-216{x}+2062\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-216,0]),K([2062,0])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-216,0]),Polrev([2062,0])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-216,0],K![2062,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((51a)\) = \((a)\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5202 \) = \(2\cdot3\cdot3\cdot17\cdot17\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1228691592)\) = \((a)^{6}\cdot(-a-1)^{12}\cdot(a-1)^{12}\cdot(-2a+3)^{2}\cdot(2a+3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1509683028251494464 \) = \(2^{6}\cdot3^{12}\cdot3^{12}\cdot17^{2}\cdot17^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1107111813625}{1228691592} \)
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 $\left(-3 a - 10 : 3 a - 57 : 1\right)$
Height \(0.75751118294226278392144699081946636186\)
Torsion structure: \(\Z/2\Z\oplus\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-4 a + 9 : 2 a - 5 : 1\right)$ $\left(20 : 66 : 1\right)$
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: \( 0.75751118294226278392144699081946636186 \)
Period: \( 0.41586793492133917640323298011485746283 \)
Tamagawa product: \( 1152 \)  =  \(2\cdot( 2^{2} \cdot 3 )\cdot( 2^{2} \cdot 3 )\cdot2\cdot2\)
Torsion order: \(12\)
Leading coefficient: \( 3.5640966225938135296595654391212010567 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-a-1)\) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((a-1)\) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\((-2a+3)\) \(17\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((2a+3)\) \(17\) \(2\) \(I_{2}\) 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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 5202.5-d consists of curves linked by isogenies of degrees dividing 12.

Base change

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

Base field Curve
\(\Q\) 102.b3
\(\Q\) 3264.w3