Properties

Label 2.2.21.1-324.1-b3
Base field \(\Q(\sqrt{21}) \)
Conductor \((18)\)
Conductor norm \( 324 \)
CM no
Base change yes: 54.a3,2646.bd3
Q-curve yes
Torsion order \( 3 \)
Rank \( 0 \)

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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

\(y^2+axy+y=x^{3}+\left(-a+1\right)x^{2}+\left(-8a+21\right)x-18a+50\)
sage: E = EllipticCurve(K, [a, -a + 1, 1, -8*a + 21, -18*a + 50])
 
gp: E = ellinit([a, -a + 1, 1, -8*a + 21, -18*a + 50],K)
 
magma: E := ChangeRing(EllipticCurve([a, -a + 1, 1, -8*a + 21, -18*a + 50]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((18)\) = \( \left(2\right) \cdot \left(-a + 2\right)^{4} \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 324 \) = \( 3^{4} \cdot 4 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((216)\) = \( \left(2\right)^{3} \cdot \left(-a + 2\right)^{6} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 46656 \) = \( 3^{6} \cdot 4^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{9261}{8} \)
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: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(0 : -2 a + 5 : 1\right)$
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: \( 13.2895953592672 \)
Tamagawa product: \( 9 \)  =  \(3\cdot3\)
Torsion order: \(3\)
Leading coefficient: \(2.90002746138933\)
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(3\) \(3\) \(IV\) Additive \(-1\) \(4\) \(6\) \(0\)
\( \left(2\right) \) \(4\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 324.1-b consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of elliptic curves 54.a3, 2646.bd3, defined over \(\Q\), so it is also a \(\Q\)-curve.