Abelian variety isogeny class 2.729.adx_fxz over $\F_{3^{6}}$

• Label: 2.729.adx_fxz
• Base field: $\F_{3^{6}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{6}}$
Dimension:	$2$
L-polynomial:	$1 - 101 x + 4003 x^{2} - 73629 x^{3} + 531441 x^{4}$
Frobenius angles:	$\pm0.0674750356186$, $\pm0.148770162783$
Angle rank:	$2$ (numerical)
Number field:	4.0.33694605.3
Galois group:	$D_{4}$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$461715$	$281265235125$	$150079806212196555$	$79766332828306001863125$	$42391159482657468191618569200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$629$	$529247$	$387382211$	$282429146123$	$205891137959114$	$150094635799985567$	$109418989150439827991$	$79766443077409976868083$	$58149737003052494972682929$	$42391158275216434300365662702$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3^{6}}$

The endomorphism algebra of this simple isogeny class is 4.0.33694605.3.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.729.dx_fxz	$2$	(not in LMFDB)