# Properties

 Label 2.2.8.1-17.1-a4 Base field $$\Q(\sqrt{2})$$ Conductor $$(-3a-1)$$ Conductor norm $$17$$ CM no Base change no Q-curve no Torsion order $$6$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{2})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-2, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);

## Weierstrass equation

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

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-3a-1)$$ = $$(-3a-1)$$ 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: $$(3a+1)$$ = $$(-3a-1)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$17$$ = $$17$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{3690752}{17} a + \frac{5287232}{17}$$ 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: $$0$$ Torsion structure: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-1 : -1 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$44.4632310427098$$ Tamagawa product: $$1$$ Torsion order: $$6$$ Leading coefficient: $$0.436670169218948$$ 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$$(j)_{-}$$
$$(-3a-1)$$ $$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## 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.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 17.1-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

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