Properties

Base field \(\Q(\sqrt{21}) \)
Label 2.2.21.1-336.1-d1
Conductor \((-8 a + 4)\)
Conductor norm \( 336 \)
CM no
base-change yes: 588.c2,252.b2
Q-curve yes
Torsion order \( 6 \)
Rank not available

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: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 5)
 
gp: K = nfinit(a^2 - a - 5);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);
 

Weierstrass equation

\( y^2 = x^{3} + a x^{2} + \left(567 a - 1585\right) x - 11818 a + 32988 \)
sage: E = EllipticCurve(K, [0, a, 0, 567*a - 1585, -11818*a + 32988])
 
gp: E = ellinit([0, a, 0, 567*a - 1585, -11818*a + 32988],K)
 
magma: E := ChangeRing(EllipticCurve([0, a, 0, 567*a - 1585, -11818*a + 32988]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-8 a + 4)\) = \( \left(2\right)^{2} \cdot \left(-a + 2\right) \cdot \left(a + 3\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 336 \) = \( 3 \cdot 4^{2} \cdot 7 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((5647152)\) = \( \left(2\right)^{4} \cdot \left(-a + 2\right)^{2} \cdot \left(a + 3\right)^{12} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 31890325711104 \) = \( 3^{2} \cdot 4^{4} \cdot 7^{12} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{10061824000}{352947} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(-9 a + 26 : 28 a - 77 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

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\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(a + 3\right) \) \(7\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\( \left(2\right) \) \(4\) \(3\) \(IV\) Additive \(1\) \(2\) \(4\) \(0\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 336.1-d consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is the base-change of elliptic curves 588.c2, 252.b2, defined over \(\Q\), so it is also a \(\Q\)-curve.