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

• Label: 3.3.ae_i_ao
• 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:	$( 1 - 2 x + 3 x^{2} )( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$
	$1 - 4 x + 8 x^{2} - 14 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.0292466093486$, $\pm0.304086723985$, $\pm0.637420057318$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$684$	$12312$	$530784$	$14114166$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$10$	$18$	$82$	$240$	$592$	$2016$	$6562$	$19494$	$59050$

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 $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac $\times$ 2.3.ac_b 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.ac_b : \(\Q(\sqrt{-2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu^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.ac_af. The endomorphism algebra for each factor is:

  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 2.9.ac_af : \(\Q(\sqrt{-2}, \sqrt{-3})\).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak^2 $\times$ 1.27.k. The endomorphism algebra for each factor is:

  • 1.27.ak^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
  • 1.27.k : \(\Q(\sqrt{-2}) \).

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.a_a_ak	$2$	3.9.a_a_abu
3.3.a_a_k	$2$	3.9.a_a_abu
3.3.e_i_o	$2$	3.9.a_a_abu
3.3.c_f_e	$3$	(not in LMFDB)