Properties

Label 2.0.8.1-3249.5-b2
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 3249 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 5 \)
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

\({y}^2+{y}={x}^{3}+{x}^{2}+20{x}-32\)
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([20,0]),K([-32,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([1,0]),Polrev([20,0]),Polrev([-32,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![20,0],K![-32,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((57)\) = \((-a-1)\cdot(a-1)\cdot(-3a+1)\cdot(3a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 3249 \) = \(3\cdot3\cdot19\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1121931)\) = \((-a-1)^{10}\cdot(a-1)^{10}\cdot(-3a+1)\cdot(3a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1258729168761 \) = \(3^{10}\cdot3^{10}\cdot19\cdot19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{841232384}{1121931} \)
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 $\left(3 a - 1 : 3 a + 1 : 1\right)$
Height \(0.28237991291240545353098885977474859547\)
Torsion structure: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 : -5 : 1\right)$
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: \( 0.28237991291240545353098885977474859547 \)
Period: \( 1.3588291502046804185980242187002843408 \)
Tamagawa product: \( 100 \)  =  \(( 2 \cdot 5 )\cdot( 2 \cdot 5 )\cdot1\cdot1\)
Torsion order: \(5\)
Leading coefficient: \( 2.1705692396487256207273716089798046319 \)
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\((j)_{-}\)
\((-a-1)\) \(3\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)
\((a-1)\) \(3\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)
\((-3a+1)\) \(19\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((3a+1)\) \(19\) \(1\) \(I_{1}\) Non-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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 3249.5-b consists of curves linked by isogenies of degree 5.

Base change

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

Base field Curve
\(\Q\) 57.b2
\(\Q\) 3648.y2