Abelian variety isogeny class 5.3.ac_l_aq_ce_acm over $\F_{3}$

• Label: 5.3.ac_l_aq_ce_acm
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $5$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$5$
L-polynomial:	$( 1 - 2 x + 3 x^{2} )( 1 + 8 x^{2} + 32 x^{4} + 72 x^{6} + 81 x^{8} )$
	$1 - 2 x + 11 x^{2} - 16 x^{3} + 56 x^{4} - 64 x^{5} + 168 x^{6} - 144 x^{7} + 297 x^{8} - 162 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.304086723985$, $\pm0.320913276015$, $\pm0.429086723985$, $\pm0.570913276015$, $\pm0.679086723985$
Angle rank:	$1$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$5$
Slopes:	$[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$388$	$451632$	$19115596$	$4176692736$	$845060291108$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$28$	$38$	$96$	$242$	$604$	$2102$	$6664$	$19874$	$59708$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{16}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac $\times$ 4.3.a_i_a_bg 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 : \(\Q(\sqrt{-2}) \).
• 4.3.a_i_a_bg : \(\Q(\zeta_{16})\).

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

The base change of $A$ to $\F_{3^{16}}$ is 1.43046721.rsg^5 and its endomorphism algebra is $\mathrm{M}_{5}($\(\Q(\sqrt{-2}) \)$)$

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.i_bg^2. The endomorphism algebra for each factor is:

  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 2.9.i_bg^2 : $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$
• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.o $\times$ 2.81.a_bi^2. The endomorphism algebra for each factor is:

  • 1.81.o : \(\Q(\sqrt{-2}) \).
  • 2.81.a_bi^2 : $\mathrm{M}_{2}($\(\Q(\zeta_{8})\)$)$
• Endomorphism algebra over $\F_{3^{8}}$
  The base change of $A$ to $\F_{3^{8}}$ is 1.6561.abi $\times$ 1.6561.bi^4. The endomorphism algebra for each factor is:

  • 1.6561.abi : \(\Q(\sqrt{-2}) \).
  • 1.6561.bi^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$

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
5.3.c_l_q_ce_cm	$2$	(not in LMFDB)
5.3.ac_af_q_i_acm	$4$	(not in LMFDB)
5.3.c_af_aq_i_cm	$4$	(not in LMFDB)