# Properties

 Label 2.2.17.1-676.5-i4 Base field $$\Q(\sqrt{17})$$ Conductor $$(20a-42)$$ Conductor norm $$676$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$1$$

# Learn more

Show commands: 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

$${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(24a+38\right){x}+15a+23$$
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,1]),K([38,24]),K([23,15])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,1])),Pol(Vecrev([38,24])),Pol(Vecrev([23,15]))], K);

magma: E := EllipticCurve([K![1,0],K![1,0],K![1,1],K![38,24],K![23,15]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(20a-42)$$ = $$(-a+2)\cdot(-a-1)\cdot(2a+1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$676$$ = $$2\cdot2\cdot13^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-34598a-33848)$$ = $$(-a+2)^{8}\cdot(-a-1)\cdot(2a+1)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$2471326208$$ = $$2^{8}\cdot2\cdot13^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{110887}{256} a + \frac{156845}{256}$$ 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: $$1$$ Generator $\left(5 : 3 a + 15 : 1\right)$ Height $$0.247456941425810$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-\frac{1}{4} a : -\frac{3}{8} a - \frac{1}{2} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.247456941425810$$ Period: $$3.28292054400517$$ Tamagawa product: $$32$$  =  $$2^{3}\cdot1\cdot2^{2}$$ Torsion order: $$2$$ Leading coefficient: $$3.15250318775630$$ 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$$(j)_{-}$$
$$(-a+2)$$ $$2$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$(-a-1)$$ $$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(2a+1)$$ $$13$$ $$4$$ $$I_0^{*}$$ Additive $$1$$ $$2$$ $$6$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 676.5-i consists of curves linked by isogenies of degrees dividing 24.

## Base change

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