Abelian variety isogeny class 4.3.ac_ab_c_e over $\F_{3}$

• Label: 4.3.ac_ab_c_e
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$1 - 2 x - x^{2} + 2 x^{3} + 4 x^{4} + 6 x^{5} - 9 x^{6} - 54 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0787816009970$, $\pm0.204819271338$, $\pm0.587885065670$, $\pm0.871485938004$
Angle rank:	$2$ (numerical)
Number field:	8.0.1768034304.4
Galois group:	$D_4\times C_2$
Jacobians:	$0$
Isomorphism classes:	24
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$28$	$3472$	$414736$	$42886144$	$4071046588$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$4$	$20$	$80$	$282$	$802$	$2046$	$6816$	$19532$	$58804$

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

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.1768034304.4.

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

The base change of $A$ to $\F_{3^{3}}$ is 2.27.ae_ba^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.4672.2$)$

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
4.3.c_ab_ac_e	$2$	(not in LMFDB)
4.3.e_o_bg_cp	$3$	(not in LMFDB)
4.3.ae_o_abg_cp	$6$	(not in LMFDB)