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

• Label: 4.3.a_ab_a_i
• 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:	$1 - x^{2} + 8 x^{4} - 9 x^{6} + 81 x^{8}$
Frobenius angles:	$\pm0.144188681100$, $\pm0.324334414339$, $\pm0.675665585661$, $\pm0.855811318900$
Angle rank:	$2$ (numerical)
Number field:	8.0.4521217600.1
Galois group:	$C_2^2 \wr C_2$
Jacobians:	$15$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$6400$	$527360$	$64000000$	$3512856400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$8$	$28$	$112$	$244$	$722$	$2188$	$6944$	$19684$	$59928$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which 5 are hyperelliptic):

• $y^2=x^9+2 x^7+2 x^5+2 x^3+2 x$
• $y^2=x^{10}+x^9+2 x^8+2 x^6+x^4+x^2+x+2$
• $y^2=x^{10}+x^9+2 x^8+x^6+x^4+2 x^2+x$
• $y^2=x^9+x^7+2 x^5+x^3+2 x$
• $y^2=x^9+x^8+x^5+2 x^3+2 x^2+2 x$
• $x y+t^2=y^2 z+y z^2-z^3+x^2 t+x z t+y z t+z^2 t=0$
• $x y+t^2=y^2 z-y z^2+z^3+x^2 t+x z t-z^2 t=0$
• $x y+t^2=y^3+x z^2-y z^2+z^3+x^2 t+y^2 t+y z t+z^2 t=0$
• $x y+t^2=y^2 z+x z^2+y z^2+z^3+x^2 t-x y t+x z t+y z t+z^2 t=0$
• $x y+t^2=y^3+x y z+y^2 z+x z^2-y z^2-z^3+x^2 t-y^2 t-x z t+y z t+z^2 t=0$
• $x y+t^2=x^2 z-x y z+y^2 z+y z^2-z^3+y^2 t+y z t=0$
• $x^2+y^2+z t=y^2 z+y z^2+y z t+z^2 t+x t^2-y t^2=0$
• $x^2+y^2+z t=y^3-y z^2-z^3+y^2 t+x z t+z^2 t-y t^2-z t^2=0$
• $x^2+y^2+z t=y^3-y z^2-z^3+y^2 t+x z t+y z t+z^2 t-x t^2=0$
• $x^2+y^2+z t=y^3+y^2 z+y z^2+x z t+z^2 t-x t^2-z t^2+t^3=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.4521217600.1.

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

The base change of $A$ to $\F_{3^{2}}$ is 2.9.ab_i^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.67240.1$)$

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
4.3.a_b_a_i	$4$	(not in LMFDB)