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

• Label: 4.3.a_a_a_ac
• 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:	$( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )( 1 + 2 x + 2 x^{2} + 6 x^{3} + 9 x^{4} )$
	$1 - 2 x^{4} + 81 x^{8}$
Frobenius angles:	$\pm0.116139763599$, $\pm0.383860236401$, $\pm0.616139763599$, $\pm0.883860236401$
Angle rank:	$1$ (numerical)
Jacobians:	$26$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$6400$	$531920$	$40960000$	$3486722000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$10$	$28$	$74$	$244$	$730$	$2188$	$7194$	$19684$	$59050$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which 9 are hyperelliptic):

• $y^2=x^9+2 x^7+2 x^6+x^5+2 x^4+x^3+2 x^2+1$
• $y^2=2 x^9+x^7+x^6+2 x^5+x^4+2 x^3+x^2+2$
• $y^2=x^9+x^7+x^5+x^3+x$
• $y^2=x^{10}+x^9+x^8+x^7+2 x^6+x^5+2 x+2$
• $y^2=x^{10}+2 x^9+x^8+2 x^6+2 x^4+x^2+2 x$
• $y^2=x^9+2 x^7+x^5+2 x^3+x$
• $y^2=x^{10}+x^9+x^8+2 x^7+2 x^6+x^5+x^4+2 x^3+2 x^2+x+2$
• $y^2=x^9+2 x^8+x^6+2 x^5+2 x^4+x^2+x$
• $y^2=x^9+2 x^6+x^5+x^4+x$
• $x y+t^2=y^2 z-y z^2+z^3+x^2 t+x z t+y z t=0$
• $x y+t^2=y^3+y^2 z+y z^2-z^3+x^2 t+y^2 t+x z t+y z t=0$
• $x y+t^2=y^3-y^2 z+y z^2-z^3+x^2 t+x z t-z^2 t=0$
• $x y+t^2=y^2 z+x z^2+z^3+x^2 t=0$
• $x y+t^2=y^3+x z^2-y z^2-z^3+x^2 t+y^2 t+y z t+z^2 t=0$
• $x y+t^2=-y^3+y^2 z+x z^2+y z^2+z^3+x^2 t+z^2 t=0$
• $x y+t^2=-y^3+y^2 z+x z^2-z^3+x^2 t-y^2 t-z^2 t=0$
• $x y+t^2=y^3+y^2 z+x z^2+y z^2-z^3+x^2 t+x z t+z^2 t=0$
• $x y+t^2=y^3-y^2 z+x z^2-y z^2-z^3+x^2 t+y^2 t+x z t+y z t-z^2 t=0$
• $x y+t^2=y^3+x z^2+y z^2-z^3+x^2 t+x y t+y^2 t+y z t=0$
• $x y+t^2=y^3+y^2 z+x z^2+y z^2-z^3+x^2 t+x y t+y^2 t+x z t=0$
• $x y+t^2=x y z+y^2 z+y z^2+z^3+x^2 t+x z t+y z t=0$
• $x y+t^2=-y^3+x y z-y^2 z+x z^2-y z^2-z^3+x^2 t+x y t+y^2 t-x z t-y z t+z^2 t=0$
• $x y+t^2=x y^2+y^2 z+x z^2+z^3+x^2 t-y^2 t-z^2 t=0$
• $x y+t^2=x^2 z-y^2 z+z^3+x y t-y^2 t-z^2 t=0$
• $x t-y z=y^3+x^2 z-z^3+x^2 t-y^2 t+x z t+z^2 t+y t^2-z t^2-t^3=0$
• $x t-y z=x y^2+x^2 z+z^3-x y t+z^2 t+y t^2-z t^2+t^3=0$

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 $\times$ 2.3.c_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 2.3.ac_c : \(\Q(i, \sqrt{5})\).
• 2.3.c_c : \(\Q(i, \sqrt{5})\).

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.ae_i_au_bu	$2$	(not in LMFDB)
4.3.e_i_u_bu	$2$	(not in LMFDB)
4.3.ac_c_i_ar	$6$	(not in LMFDB)