# Properties

 Label 2.0.67.1-289.2-b1 Base field $$\Q(\sqrt{-67})$$ Conductor $$(17)$$ Conductor norm $$289$$ CM no Base change no Q-curve no Torsion order $$4$$ Rank not available

# Learn more

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-67})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x + 17$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([17, -1, 1]))

gp: K = nfinit(Pol(Vecrev([17, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -1, 1]);

## Weierstrass equation

$${y}^2+a{x}{y}+a{y}={x}^3-{x}^2+\left(-a+17\right){x}$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([17,-1]),K([0,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([17,-1])),Pol(Vecrev([0,0]))], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![17,-1],K![0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(17)$$ = $$(-a)\cdot(a-1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$289$$ = $$17\cdot17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-17a)$$ = $$(-a)^{2}\cdot(a-1)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4913$$ = $$17^{2}\cdot17$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{148207}{289} a - \frac{1202001}{289}$$ 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: $$0 \le r \le 1$$ Torsion structure: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(0 : -a : 1\right)$ 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: $$0 \le r \le 1$$ Regulator: not available Period: $$6.34473030369887$$ Tamagawa product: $$2$$  =  $$2\cdot1$$ Torsion order: $$4$$ Leading coefficient: $$5.44074508706850$$ Analytic order of Ш: not available

## 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$$(j)_{-}$$
$$(-a)$$ $$17$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(a-1)$$ $$17$$ $$1$$ $$I_{1}$$ 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 289.2-b consists of curves linked by isogenies of degrees dividing 4.

## Base change

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