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

• Label: 4.3.ae_i_au_bu
• 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:	$( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )^{2}$
	$1 - 4 x + 8 x^{2} - 20 x^{3} + 46 x^{4} - 60 x^{5} + 72 x^{6} - 108 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.116139763599$, $\pm0.116139763599$, $\pm0.616139763599$, $\pm0.616139763599$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$6400$	$190096$	$40960000$	$4767073936$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$10$	$0$	$74$	$320$	$730$	$2240$	$7194$	$20160$	$59050$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=2 x^{10}+x^8+x^6+x^4+x^2+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 2.3.ac_c^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(i, \sqrt{5})\)$)$

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

The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-5}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$

  The base change of $A$ to $\F_{3^{2}}$ is 2.9.a_ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(i, \sqrt{5})\)$)$

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.a_a_a_ac	$2$	(not in LMFDB)
4.3.e_i_u_bu	$2$	(not in LMFDB)
4.3.c_c_ai_ar	$3$	(not in LMFDB)