# Properties

 Label 2.2.17.1-81.1-c6 Base field $$\Q(\sqrt{17})$$ Conductor $$(9)$$ Conductor norm $$81$$ CM no Base change no Q-curve yes Torsion order $$4$$ Rank $$0$$

# Learn more about

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{17})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 4$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-4, -1, 1]))

gp: K = nfinit(Pol(Vecrev([-4, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);

## Weierstrass equation

$$y^2+xy+y=x^{3}-x^{2}+\left(-15a-35\right)x-58a-74$$
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-35,-15]),K([-74,-58])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-35,-15])),Pol(Vecrev([-74,-58]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-35,-15],K![-74,-58]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(9)$$ = $$\left(3\right)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$81$$ = $$9^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(6561)$$ = $$\left(3\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$43046721$$ = $$9^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{14326000}{9} a + \frac{22587625}{9}$$ 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: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-\frac{7}{4} a - \frac{1}{4} : \frac{7}{8} a - \frac{3}{8} : 1\right)$ $\left(-\frac{1}{4} a - 4 : \frac{1}{8} a + \frac{3}{2} : 1\right)$ 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: $$6.82841002510382$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$0.414033173360729$$ 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$$(j)_{-}$$
$$\left(3\right)$$ $$9$$ $$4$$ $$I_{2}^*$$ Additive $$1$$ $$2$$ $$8$$ $$2$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 81.1-c consists of curves linked by isogenies of degrees dividing 16.

## Base change

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