# Properties

 Label 2.2.21.1-17.1-a2 Base field $$\Q(\sqrt{21})$$ Conductor $$(-2 a + 3)$$ Conductor norm $$17$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$0$$

# Related objects

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

$$y^2+\left(a+1\right)xy+\left(a+1\right)y=x^{3}-ax^{2}+\left(-15a+33\right)x+10a-33$$
sage: E = EllipticCurve(K, [a + 1, -a, a + 1, -15*a + 33, 10*a - 33])

gp: E = ellinit([a + 1, -a, a + 1, -15*a + 33, 10*a - 33],K)

magma: E := ChangeRing(EllipticCurve([a + 1, -a, a + 1, -15*a + 33, 10*a - 33]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-2 a + 3)$$ = $$\left(-2 a + 3\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$17$$ = $$17$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(-31 a + 4)$$ = $$\left(-2 a + 3\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4913$$ = $$17^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{2723256379739}{4913} a + \frac{4878142779916}{4913}$$ 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: 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$5.01827598389182$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$1.09507759782682$$ 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$$(j)_{-}$$
$$\left(-2 a + 3\right)$$ $$17$$ $$1$$ $$I_{3}$$ Non-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
$$3$$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 17.1-a consists of curves linked by isogenies of degree 3.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.