# Properties

 Label 2.2.17.1-289.1-a7 Base field $$\Q(\sqrt{17})$$ Conductor $$(17)$$ Conductor norm $$289$$ CM no Base change no Q-curve yes Torsion order $$4$$ Rank $$0$$

# Related objects

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: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 4)

gp: K = nfinit(a^2 - a - 4);

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

## Weierstrass equation

$$y^2+xy=x^{3}-x^{2}+\left(-24667a-38542\right)x-3022719a-4720173$$
sage: E = EllipticCurve(K, [1, -1, 0, -24667*a - 38542, -3022719*a - 4720173])

gp: E = ellinit([1, -1, 0, -24667*a - 38542, -3022719*a - 4720173],K)

magma: E := ChangeRing(EllipticCurve([1, -1, 0, -24667*a - 38542, -3022719*a - 4720173]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(17)$$ = $$\left(-2 a + 1\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$289$$ = $$17^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(83521)$$ = $$\left(-2 a + 1\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$6975757441$$ = $$17^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{82483294977}{17}$$ 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: $\left(-44 a - \frac{285}{4} : 22 a + \frac{285}{8} : 1\right)$ $\left(-46 a - 66 : 23 a + 33 : 1\right)$ 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: $$0.580498289681639$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$2.25266424832734$$ Analytic order of Ш: $$64$$ (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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$17$$ $$4$$ $$I_{2}^*$$ 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 289.1-a 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.