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

• Label: 4.3.c_a_ae_af
• 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:	$1 + 2 x - 4 x^{3} - 5 x^{4} - 12 x^{5} + 54 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0988888664344$, $\pm0.455899292071$, $\pm0.765555533101$, $\pm0.877434041262$
Angle rank:	$2$ (numerical)
Number field:	8.0.56070144.2
Galois group:	$D_4\times C_2$
Jacobians:	$0$
Isomorphism classes:	42

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$117$	$4329$	$492804$	$41251041$	$2764026837$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$6$	$24$	$78$	$186$	$834$	$2274$	$6630$	$19680$	$59766$

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 8.0.56070144.2.

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

The base change of $A$ to $\F_{3^{3}}$ is 2.27.ac_bc^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.7488.1$)$

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.ac_a_e_af	$2$	(not in LMFDB)
4.3.ae_m_abc_cg	$3$	(not in LMFDB)
4.3.ag_u_abw_dn	$4$	(not in LMFDB)