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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$( 1 - 4 x + 5 x^{2} )( 1 - 4 x + 12 x^{2} - 20 x^{3} + 25 x^{4} )$
	$1 - 8 x + 33 x^{2} - 88 x^{3} + 165 x^{4} - 200 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.223508181938$, $\pm0.458185759261$
Angle rank:	$3$ (numerical)
Jacobians:	$0$
Isomorphism classes:	6

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$28$	$17360$	$2256268$	$251095040$	$31627095388$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$28$	$142$	$644$	$3238$	$16156$	$79014$	$390076$	$1949182$	$9763068$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae $\times$ 2.5.ae_m and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.ae : \(\Q(\sqrt{-1}) \).
• 2.5.ae_m : 4.0.10496.2.

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
3.5.a_b_ai	$2$	3.25.c_l_ho
3.5.a_b_i	$2$	3.25.c_l_ho
3.5.i_bh_dk	$2$	3.25.c_l_ho