# Properties

 Label 2.2.17.1-676.5-i7 Base field $$\Q(\sqrt{17})$$ Conductor $$(20a-42)$$ Conductor norm $$676$$ CM no Base change no Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

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

$${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-956a-1508\right){x}+22281a+34805$$
sage: E = EllipticCurve([K([1,0]),K([1,-1]),K([1,0]),K([-1508,-956]),K([34805,22281])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,-1])),Pol(Vecrev([1,0])),Pol(Vecrev([-1508,-956])),Pol(Vecrev([34805,22281]))], K);

magma: E := EllipticCurve([K![1,0],K![1,-1],K![1,0],K![-1508,-956],K![34805,22281]]);

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: $$(189376a+1096000)$$ = $$(-a+2)^{6}\cdot(-a-1)^{12}\cdot(2a+1)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1265319018496$$ = $$2^{6}\cdot2^{12}\cdot13^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{203862548967}{4096} a + \frac{130566616997}{1024}$$ 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 $\left(6 a + 11 : -20 a - 33 : 1\right)$ Height $$0.329942588567746$$ 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: $\left(-\frac{71}{4} a - \frac{105}{4} : \frac{71}{8} a + \frac{101}{8} : 1\right)$ $\left(\frac{27}{4} a + 12 : -\frac{27}{8} a - \frac{13}{2} : 1\right)$ 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: $$0.329942588567746$$ Period: $$6.56584108801035$$ Tamagawa product: $$48$$  =  $$( 2 \cdot 3 )\cdot2\cdot2^{2}$$ Torsion order: $$4$$ 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$$(j)_{-}$$
$$(-a+2)$$ $$2$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$(-a-1)$$ $$2$$ $$2$$ $$I_{12}$$ Non-split multiplicative $$1$$ $$1$$ $$12$$ $$12$$
$$(2a+1)$$ $$13$$ $$4$$ $$I_0^{*}$$ 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
$$2$$ 2Cs
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
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.