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

• Label: 4.3.ac_a_e_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: yes

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:	$\pm0.122565958738$, $\pm0.234444466899$, $\pm0.544100707929$, $\pm0.901111133566$
Angle rank:	$2$ (numerical)
Number field:	8.0.56070144.2
Galois group:	$D_4\times C_2$
Jacobians:	$1$
Cyclic group of points:	yes

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$37$	$4329$	$662596$	$41251041$	$4452218917$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$6$	$32$	$78$	$302$	$834$	$2102$	$6630$	$19688$	$59766$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $x y+t^2=-y^3-y^2 z+x z^2-y z^2+z^3+x^2 t-y^2 t+x z t-y z t=0$

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.c_bc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-8 -2 \sqrt{3}})\)$)$

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_a_ae_af	$2$	(not in LMFDB)
4.3.e_m_bc_cg	$3$	(not in LMFDB)
4.3.ag_u_abw_dn	$4$	(not in LMFDB)