Abelian variety isogeny class 4.5.aq_em_atc_cao over $\F_{5}$

• Label: 4.5.aq_em_atc_cao
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{5}$
Dimension:	$4$
L-polynomial:	$( 1 - 4 x + 5 x^{2} )^{4}$
	$1 - 16 x + 116 x^{2} - 496 x^{3} + 1366 x^{4} - 2480 x^{5} + 2900 x^{6} - 2000 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.147583617650$
Angle rank:	$1$ (numerical)

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$160000$	$221533456$	$167772160000$	$105119989862416$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-10$	$2$	$110$	$682$	$3430$	$16562$	$80350$	$394842$	$1958870$	$9767522$

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^4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-1}) \)$)$

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
4.5.ai_u_i_aec	$2$	(not in LMFDB)
4.5.a_am_a_di	$2$	(not in LMFDB)
4.5.i_u_ai_aec	$2$	(not in LMFDB)
4.5.q_em_tc_cao	$2$	(not in LMFDB)
4.5.ae_f_ae_q	$3$	(not in LMFDB)
4.5.i_bm_ey_mt	$3$	(not in LMFDB)