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

• Label: 3.3.ag_t_abo
• 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:	$( 1 - 2 x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
	$1 - 6 x + 19 x^{2} - 40 x^{3} + 57 x^{4} - 54 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.0540867239847$, $\pm0.304086723985$, $\pm0.445913276015$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4$	$816$	$23788$	$443904$	$11926244$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$12$	$34$	$68$	$198$	$684$	$2154$	$6460$	$19582$	$59532$

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 $\F_{3^{8}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac $\times$ 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.ac : \(\Q(\sqrt{-2}) \).
• 2.3.ae_i : \(\Q(\zeta_{8})\).

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

The base change of $A$ to $\F_{3^{8}}$ is 1.6561.abi^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-2}) \)$)$

Remainder of endomorphism lattice by field

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

  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 2.9.a_ao : \(\Q(\zeta_{8})\).
• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao^2 $\times$ 1.81.o. The endomorphism algebra for each factor is:

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

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.ac_d_ai	$2$	3.9.c_af_abc
3.3.c_d_i	$2$	3.9.c_af_abc
3.3.g_t_bo	$2$	3.9.c_af_abc