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

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

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$( 1 + x + 2 x^{2} + 3 x^{3} + 9 x^{4} )^{2}$
	$1 + 2 x + 5 x^{2} + 10 x^{3} + 28 x^{4} + 30 x^{5} + 45 x^{6} + 54 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.351145371037$, $\pm0.351145371037$, $\pm0.764917483542$, $\pm0.764917483542$
Angle rank:	$2$ (numerical)
Jacobians:	$3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$256$	$16384$	$692224$	$75759616$	$2320926976$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$16$	$36$	$128$	$146$	$658$	$2246$	$6528$	$20700$	$58576$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which 1 is hyperelliptic):

• $y^2=2 x^9+2 x^6+2 x^5+2 x^4+2 x$
• $x y+t^2=x^2 z-y^2 z-z^3+y^2 t+z^2 t=0$
• $x^2+y^2+z t=y^2 z+y z^2+y z t-z^2 t+x t^2+z t^2=0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 2.3.b_c^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.3757.1$)$

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_f_ak_bc	$2$	(not in LMFDB)
4.3.a_d_a_q	$2$	(not in LMFDB)
4.3.ab_ab_e_ai	$3$	(not in LMFDB)