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

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

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$1 - x - 2 x^{2} + 5 x^{3} + x^{4} + 15 x^{5} - 18 x^{6} - 27 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.193214749339$, $\pm0.206785250661$, $\pm0.606785250661$, $\pm0.993214749339$
Angle rank:	$1$ (numerical)
Number field:	8.0.228765625.1
Galois group:	$C_4\times C_2$
Isomorphism classes:	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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$55$	$3905$	$717805$	$46489025$	$5719140625$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$5$	$36$	$89$	$368$	$740$	$2271$	$6449$	$19548$	$57150$

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 $\F_{3^{5}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.228765625.1.

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

The base change of $A$ to $\F_{3^{5}}$ is 1.243.bf^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-11}) \)$)$

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.b_ac_af_b	$2$	(not in LMFDB)
4.3.e_s_bo_dn	$5$	(not in LMFDB)
4.3.ae_s_abo_dn	$10$	(not in LMFDB)