Properties

Label 2.2.204.1-16.1-b1
Base field \(\Q(\sqrt{51}) \)
Conductor norm \( 16 \)
CM no
Base change no
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{51}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 51 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-51, 0, 1]))
 
gp: K = nfinit(Polrev([-51, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(w+1\right){y}={x}^3-w{x}^2+16{x}-2w-13\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([1,1]),K([16,0]),K([-13,-2])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([1,1]),Polrev([16,0]),Polrev([-13,-2])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![1,1],K![16,0],K![-13,-2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((4)\) = \((a-7)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(2^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4)\) = \((a-7)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 16 \) = \(2^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -27648 \)
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 $\left(\frac{5}{3} : \frac{1}{9} a - 4 : 1\right)$
Height \(2.0545169100349089327594668562540456975\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 2.0545169100349089327594668562540456975 \)
Period: \( 18.519241893537979294616804678258506802 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 5.3277990547703476125651411140692571089 \)
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\((j)_{-}\)
\((a-7)\) \(2\) \(1\) \(II\) Additive \(-1\) \(4\) \(4\) \(0\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 16.1-b consists of this curve only.

Base change

This elliptic curve is a \(\Q\)-curve.

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