Abelian variety isogeny class 3.3.af_r_abi over $\F_{3}$

• Label: 3.3.af_r_abi
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$3$
L-polynomial:	$( 1 - x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$
	$1 - 5 x + 17 x^{2} - 34 x^{3} + 51 x^{4} - 45 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.304086723985$, $\pm0.304086723985$, $\pm0.406785250661$
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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$2160$	$51984$	$691200$	$12474132$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$19$	$56$	$103$	$209$	$628$	$2099$	$6607$	$19928$	$59539$

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 $\F_{3}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac^2 $\times$ 1.3.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.3.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.3.ab : \(\Q(\sqrt{-11}) \).

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
3.3.ad_j_ao	$2$	3.9.j_bz_ha
3.3.ab_f_ac	$2$	3.9.j_bz_ha
3.3.b_f_c	$2$	3.9.j_bz_ha
3.3.d_j_o	$2$	3.9.j_bz_ha
3.3.f_r_bi	$2$	3.9.j_bz_ha
3.3.b_c_l	$3$	(not in LMFDB)