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

• Label: 3.5.ah_bb_acs
• Base field: $\F_{5}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 + 5 x^{2} )$
	$1 - 7 x + 27 x^{2} - 70 x^{3} + 135 x^{4} - 175 x^{5} + 125 x^{6}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.265942140215$, $\pm0.5$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	60

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

Newton polygon

$p$-rank:	$2$
Slopes:	$[0, 0, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$19440$	$2213568$	$248832000$	$31860085716$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$31$	$140$	$639$	$3259$	$16036$	$78175$	$389279$	$1953644$	$9775351$

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 $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae $\times$ 1.5.ad $\times$ 1.5.a 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 : \(\Q(\sqrt{-1}) \).
• 1.5.ad : \(\Q(\sqrt{-11}) \).
• 1.5.a : \(\Q(\sqrt{-5}) \).

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.b $\times$ 1.25.k. The endomorphism algebra for each factor is:

• 1.25.ag : \(\Q(\sqrt{-1}) \).
• 1.25.b : \(\Q(\sqrt{-11}) \).
• 1.25.k : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

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.ab_d_ak	$2$	3.25.f_t_hi
3.5.b_d_k	$2$	3.25.f_t_hi
3.5.h_bb_cs	$2$	3.25.f_t_hi