Properties

Label 2.2.24.1-784.1-l1
Base field \(\Q(\sqrt{6}) \)
Conductor norm \( 784 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{6}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 6 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
 
gp: K = nfinit(Polrev([-6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-a-2\right){x}-2a-5\)
sage: E = EllipticCurve([K([0,1]),K([-1,1]),K([0,0]),K([-2,-1]),K([-5,-2])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([-1,1]),Polrev([0,0]),Polrev([-2,-1]),Polrev([-5,-2])], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,1],K![0,0],K![-2,-1],K![-5,-2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((28)\) = \((-a+2)^{4}\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 784 \) = \(2^{4}\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((448a-896)\) = \((-a+2)^{13}\cdot(7)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -401408 \) = \(-2^{13}\cdot49\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{14841}{14} a - \frac{2603}{7} \)
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 $\left(-a : 1 : 1\right)$
Height \(0.17717098994184600921311402413353414278\)
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: \( 0.17717098994184600921311402413353414278 \)
Period: \( 7.4649903511912275912240599718918273847 \)
Tamagawa product: \( 4 \)  =  \(2^{2}\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.1597636557957197494067942925987365838 \)
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\((j)_{-}\)
\((-a+2)\) \(2\) \(4\) \(I_{5}^{*}\) Additive \(1\) \(4\) \(13\) \(1\)
\((7)\) \(49\) \(1\) \(I_{1}\) 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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 784.1-l consists of curves linked by isogenies of degree 3.

Base change

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

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