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

• Label: 3.3.af_p_abg
• 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:	$( 1 - x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
	$1 - 5 x + 15 x^{2} - 32 x^{3} + 45 x^{4} - 45 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.0540867239847$, $\pm0.406785250661$, $\pm0.445913276015$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$1020$	$22536$	$346800$	$10497066$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$15$	$32$	$47$	$169$	$720$	$2323$	$6607$	$19256$	$58575$

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 $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ab $\times$ 2.3.ae_i 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.ab : \(\Q(\sqrt{-11}) \).
• 2.3.ae_i : \(\Q(\zeta_{8})\).

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao^2 $\times$ 1.81.ah. The endomorphism algebra for each factor is:

• 1.81.ao^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.81.ah : \(\Q(\sqrt{-11}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$
  The base change of $A$ to $\F_{3^{2}}$ is 1.9.f $\times$ 2.9.a_ao. The endomorphism algebra for each factor is:

  • 1.9.f : \(\Q(\sqrt{-11}) \).
  • 2.9.a_ao : \(\Q(\zeta_{8})\).

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_h_aq	$2$	3.9.f_af_acs
3.3.d_h_q	$2$	3.9.f_af_acs
3.3.f_p_bg	$2$	3.9.f_af_acs