# Properties

 Label 4.4.4913.1-17.1-a8 Base field 4.4.4913.1 Conductor $$(1/2a^3-2a-3/2)$$ Conductor norm $$17$$ CM no Base change yes: 17.a4,289.a4 Q-curve yes Torsion order $$16$$ Rank $$1$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field4.4.4913.1

Generator $$a$$, with minimal polynomial $$x^{4} - x^{3} - 6 x^{2} + x + 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 1, -6, -1, 1]))

gp: K = nfinit(Pol(Vecrev([1, 1, -6, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -1, 1]);

## Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}$$
sage: E = EllipticCurve([K([1,0,0,0]),K([-1,0,0,0]),K([1,0,0,0]),K([-1,0,0,0]),K([0,0,0,0])])

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(1/2a^3-2a-3/2)$$ = $$(1/2a^3-2a-3/2)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$17$$ = $$17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(17)$$ = $$(1/2a^3-2a-3/2)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$83521$$ = $$17^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{35937}{17}$$ 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: $$1$$ Generator $\left(a^{3} - a^{2} - 5 a + 2 : -2 a^{3} + 2 a^{2} + 10 a - 5 : 1\right)$ Height $$0.758814044432583$$ Torsion structure: $$\Z/2\Z\times\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{5}{4} a : -\frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{5}{8} a - \frac{1}{2} : 1\right)$ $\left(\frac{3}{2} a^{3} - 2 a^{2} - 8 a + \frac{9}{2} : 3 a^{3} - 4 a^{2} - 17 a + 8 : 1\right)$ 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: $$0.758814044432583$$ Period: $$1466.52940633504$$ Tamagawa product: $$4$$ Torsion order: $$16$$ Leading coefficient: $$0.992276649128961$$ 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$$(j)_{-}$$
$$(1/2a^3-2a-3/2)$$ $$17$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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 17.1-a consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of 17.a4, 289.a4, defined over $$\Q$$, so it is also a $$\Q$$-curve.