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

• Label: 3.5.ah_bc_acy
• 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:	$( 1 - 2 x + 5 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4} )$
	$1 - 7 x + 28 x^{2} - 76 x^{3} + 140 x^{4} - 175 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.0878807261908$, $\pm0.352416382350$, $\pm0.450170915301$
Angle rank:	$3$ (numerical)
Jacobians:	$0$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$19872$	$2233764$	$226143360$	$28799113536$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$33$	$143$	$577$	$2944$	$15561$	$78763$	$392033$	$1954979$	$9770988$

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.ac $\times$ 2.5.af_n 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.ac : \(\Q(\sqrt{-1}) \).
• 2.5.af_n : 4.0.4901.1.

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.ad_i_ay	$2$	3.25.h_a_afg
3.5.d_i_y	$2$	3.25.h_a_afg
3.5.h_bc_cy	$2$	3.25.h_a_afg