Abelian variety isogeny class 2.167.aby_bkx over $\F_{167}$

• Label: 2.167.aby_bkx
• Base field: $\F_{167}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{167}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x + 167 x^{2} )^{2}$
	$1 - 50 x + 959 x^{2} - 8350 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.0816525061160$, $\pm0.0816525061160$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20449$	$761704801$	$21663104244496$	$604922158057921561$	$16871874701989750017289$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$118$	$27308$	$4651264$	$777738516$	$129891575858$	$21691961006222$	$3622557640268174$	$604967118401553508$	$101029508559556363648$	$16871927925364715569628$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=119x^6+118x^5+20x^4+133x^3+124x^2+96x+39$
• $y^2=139x^6+154x^5+56x^4+45x^3+56x^2+154x+139$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{167}$.

Endomorphism algebra over $\F_{167}$

The isogeny class factors as 1.167.az^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.167.a_alf	$2$	(not in LMFDB)
2.167.by_bkx	$2$	(not in LMFDB)
2.167.z_rq	$3$	(not in LMFDB)