Abelian variety isogeny class 2.67.abf_ok over $\F_{67}$

• Label: 2.67.abf_ok
• Base field: $\F_{67}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{67}$
Dimension:	$2$
L-polynomial:	$( 1 - 16 x + 67 x^{2} )( 1 - 15 x + 67 x^{2} )$
	$1 - 31 x + 374 x^{2} - 2077 x^{3} + 4489 x^{4}$
Frobenius angles:	$\pm0.0678686046652$, $\pm0.131184157393$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2756$	$19214832$	$90086353136$	$405962746547904$	$1822836508553125676$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$37$	$4277$	$299524$	$20145913$	$1350124147$	$90458681222$	$6060715982617$	$406067722794481$	$27206534787005308$	$1822837807494023957$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{67}$.

Endomorphism algebra over $\F_{67}$

The isogeny class factors as 1.67.aq $\times$ 1.67.ap and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.67.aq : \(\Q(\sqrt{-3}) \).
• 1.67.ap : \(\Q(\sqrt{-43}) \).

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
2.67.ab_aec	$2$	(not in LMFDB)
2.67.b_aec	$2$	(not in LMFDB)
2.67.bf_ok	$2$	(not in LMFDB)
2.67.ak_ch	$3$	(not in LMFDB)
2.67.ae_abf	$3$	(not in LMFDB)