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

• Label: 4.5.ao_dp_aoo_bnk
• 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:	$( 1 - 4 x + 5 x^{2} )^{2}( 1 - 3 x + 5 x^{2} )^{2}$
	$1 - 14 x + 93 x^{2} - 378 x^{3} + 1024 x^{4} - 1890 x^{5} + 2325 x^{6} - 1750 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.147583617650$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.265942140215$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$291600$	$308634624$	$186624000000$	$103876170945156$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-8$	$16$	$154$	$752$	$3392$	$15946$	$78224$	$390432$	$1954162$	$9772576$

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^2 $\times$ 1.5.ad^2 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 1.5.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$

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_bb_abw_cy	$2$	(not in LMFDB)
4.5.ag_n_g_acm	$2$	(not in LMFDB)
4.5.ac_ad_ag_cm	$2$	(not in LMFDB)
4.5.a_af_a_bs	$2$	(not in LMFDB)
4.5.c_ad_g_cm	$2$	(not in LMFDB)
4.5.g_n_ag_acm	$2$	(not in LMFDB)
4.5.i_bb_bw_cy	$2$	(not in LMFDB)
4.5.o_dp_oo_bnk	$2$	(not in LMFDB)
4.5.af_g_v_adi	$3$	(not in LMFDB)
4.5.ac_g_a_t	$3$	(not in LMFDB)
4.5.h_bb_dg_ig	$3$	(not in LMFDB)