# Properties

 Label 2.2.17.1-144.4-a1 Base field $$\Q(\sqrt{17})$$ Conductor $$(3 a + 12)$$ Conductor norm $$144$$ CM no Base change no Q-curve yes Torsion order $$1$$ 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: 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

$$y^2+ay=x^{3}+\left(-a-1\right)x^{2}+\left(1938a-4965\right)x-66328a+169901$$
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,1]),K([-4965,1938]),K([169901,-66328])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([-4965,1938])),Pol(Vecrev([169901,-66328]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,1],K![-4965,1938],K![169901,-66328]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(3 a + 12)$$ = $$\left(-a + 2\right)^{4} \cdot \left(3\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$144$$ = $$2^{4} \cdot 9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(8991 a - 14580)$$ = $$\left(-a + 2\right)^{12} \cdot \left(3\right)^{5}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$241864704$$ = $$2^{12} \cdot 9^{5}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{13549359104}{243}$$ 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: $$2.29528613793852$$ Tamagawa product: $$5$$  =  $$1\cdot5$$ Torsion order: $$1$$ Leading coefficient: $$2.78344329051074$$ 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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$2$$ $$1$$ $$II^*$$ Additive $$-1$$ $$4$$ $$12$$ $$0$$
$$\left(3\right)$$ $$9$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5B.4.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 144.4-a consists of curves linked by isogenies of degree 5.

## Base change

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