# Properties

 Base field 4.4.4913.1 Label 4.4.4913.1-17.1-a8 Conductor $$(17,\frac{1}{2} a^{3} - a^{2} - a + \frac{3}{2})$$ Conductor norm $$17$$ CM no base-change yes: 17.a4,289.a4 Q-curve yes Torsion order $$16$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field 4.4.4913.1

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 6*x^2 + x + 1)
gp (2.8): K = nfinit(a^4 - a^3 - 6*a^2 + a + 1);

## Weierstrass equation

$$y^2 + x y + y = x^{3} - x^{2} - x$$
magma: E := ChangeRing(EllipticCurve([1, -1, 1, -1, 0]),K);
sage: E = EllipticCurve(K, [1, -1, 1, -1, 0])
gp (2.8): E = ellinit([1, -1, 1, -1, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(17,\frac{1}{2} a^{3} - a^{2} - a + \frac{3}{2})$$ = $$\left(\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$17$$ = $$17$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(17,17 a,\frac{17}{2} a^{3} - 51 a - \frac{51}{2},-\frac{17}{2} a^{3} + 17 a^{2} + 34 a - \frac{51}{2})$$ = $$\left(\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\right)^{4}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$83521$$ = $$17^{4}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{35937}{17}$$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E); sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/8\Z$$ magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1] $\left(\frac{3}{2} a^{3} - 2 a^{2} - 8 a + \frac{9}{2} : 3 a^{3} - 4 a^{2} - 17 a + 8 : 1\right)$,$\left(\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{5}{4} a : -\frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{5}{8} a - \frac{1}{2} : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(\frac{1}{2} a^{3} - 2 a - \frac{3}{2}\right)$$ $$17$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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 elliptic curves 17.a4, 289.a4, defined over $$\Q$$, so it is also a $$\Q$$-curve.