Properties

Label 2.2.21.1-1792.1-bp4
Base field \(\Q(\sqrt{21}) \)
Conductor norm \( 1792 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 2 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{21}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 5 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))
 
gp: K = nfinit(Polrev([-5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}-a{x}^{2}+\left(2842a-7955\right){x}+104295a-291134\)
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([-7955,2842]),K([-291134,104295])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([-7955,2842]),Polrev([-291134,104295])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![-7955,2842],K![-291134,104295]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((16a+48)\) = \((2)^{4}\cdot(a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1792 \) = \(4^{4}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3855122432)\) = \((2)^{15}\cdot(a+3)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 14861968965709594624 \) = \(4^{15}\cdot7^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{4956477625}{941192} \)
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: \(2\)
Generators $\left(12 a - 35 : 64 a - 176 : 1\right)$ $\left(-\frac{262}{9} a + \frac{265}{3} : \frac{1372}{9} a - \frac{11662}{27} : 1\right)$
Heights \(1.0523342629215665640574094180261083997\) \(2.3381381113430793900703914878457978149\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(9 a - 26 : 0 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 2.4605028460090431843192578013365464453 \)
Period: \( 0.98142898671773576300769093258739825766 \)
Tamagawa product: \( 8 \)  =  \(2^{2}\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 4.2156358794165526046806902820394368671 \)
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\((j)_{-}\)
\((2)\) \(4\) \(4\) \(I_{7}^{*}\) Additive \(1\) \(4\) \(15\) \(3\)
\((a+3)\) \(7\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 1792.1-bp consists of curves linked by isogenies of degrees dividing 18.

Base change

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

Base field Curve
\(\Q\) 784.b3
\(\Q\) 1008.h3