Properties

Label 2.2.21.1-84.1-a1
Base field \(\Q(\sqrt{21}) \)
Conductor \((-4a+2)\)
Conductor norm \( 84 \)
CM no
Base change yes: 294.g5,126.a5
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / 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(Pol(Vecrev([-5, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(22a-52\right){x}+166a-457\)
sage: E = EllipticCurve([K([0,1]),K([1,1]),K([1,0]),K([-52,22]),K([-457,166])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,1])),Pol(Vecrev([1,0])),Pol(Vecrev([-52,22])),Pol(Vecrev([-457,166]))], K);
 
magma: E := EllipticCurve([K![0,1],K![1,1],K![1,0],K![-52,22],K![-457,166]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-4a+2)\) = \((-a+2)\cdot(2)\cdot(a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 84 \) = \(3\cdot4\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-16128)\) = \((-a+2)^{4}\cdot(2)^{8}\cdot(a+3)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 260112384 \) = \(3^{4}\cdot4^{8}\cdot7^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{7189057}{16128} \)
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: \(0\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-7 a + 19 : 30 a - 85 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2.48688727658968 \)
Tamagawa product: \( 64 \)  =  \(2^{2}\cdot2^{3}\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 2.17073317900853 \)
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\((j)_{-}\)
\((-a+2)\) \(3\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((2)\) \(4\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((a+3)\) \(7\) \(2\) \(I_{2}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 84.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 294.g5, 126.a5, defined over \(\Q\), so it is also a \(\Q\)-curve.