Abelian variety isogeny class 3.5.a_h_ae over $\F_{5}$

• Label: 3.5.a_h_ae
• Base field: $\F_{5}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$1 + 7 x^{2} - 4 x^{3} + 35 x^{4} + 125 x^{6}$
Frobenius angles:	$\pm0.260976308728$, $\pm0.536902371779$, $\pm0.691755156756$
Angle rank:	$3$ (numerical)
Number field:	6.0.29215664.3
Galois group:	$S_4\times C_2$
Jacobians:	$11$
Isomorphism classes:	28

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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$164$	$28208$	$1769396$	$260416256$	$31932984644$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$40$	$114$	$668$	$3266$	$15544$	$77734$	$388540$	$1952142$	$9776280$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which 5 are hyperelliptic):

• $y^2=x^7+2 x^5+x^4+4 x^3+x+2$
• $y^2=x^8+x^5+3 x^4+4 x^3+x+3$
• $y^2=x^8+x^5+4 x^4+4 x^3+4 x^2+4 x+4$
• $y^2=x^8+x^6+2 x^3+x^2+3$
• $y^2=x^8+x^6+2 x^4+x^3+x^2+3$
• $4 x^3 y+3 x^3 z+2 x^2 y^2+x^2 y z+x y^2 z+x z^3+y^4=0$
• $3 x^3 y+3 x^3 z+2 x^2 y z+x y^2 z+x z^3+y^4=0$
• $x^3 y+2 x^3 z+x^2 y^2+2 x^2 y z+x y^2 z+x z^3+y^4=0$
• $4 x^4+2 x^3 y+4 x^3 z+3 x^2 y^2+x^2 y z+2 x y^2 z+x z^3+y^4=0$
• $4 x^4+x^3 y+4 x^3 z+2 x^2 y z+x y z^2+x z^3+y^3 z=0$
• $x^4+x^3 y+x^3 z+3 x^2 y^2+2 x^2 y z+3 x^2 z^2+x y z^2+x z^3+y^3 z=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5}$.

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 6.0.29215664.3.

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
3.5.a_h_e	$2$	3.25.o_ep_bbw