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

• Label: 4.5.al_cd_agr_qe
• Base field: $\F_{5}$
• Dimension: $4$
• $p$-rank: $3$
• Ordinary: no
• 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:	$( 1 - 4 x + 5 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 15 x^{3} + 25 x^{4} )$
	$1 - 11 x + 55 x^{2} - 173 x^{3} + 420 x^{4} - 865 x^{5} + 1375 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.113143297209$, $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.585923223955$
Angle rank:	$3$ (numerical)

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$52$	$254800$	$188267716$	$152635392000$	$104445396370432$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$15$	$91$	$623$	$3410$	$16215$	$79151$	$394623$	$1960615$	$9766650$

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 $\times$ 2.5.ad_f 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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 2.5.ad_f : 4.0.37845.1.

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.af_h_n_aci	$2$	(not in LMFDB)
4.5.ad_ab_d_u	$2$	(not in LMFDB)
4.5.d_ab_ad_u	$2$	(not in LMFDB)
4.5.f_h_an_aci	$2$	(not in LMFDB)
4.5.l_cd_gr_qe	$2$	(not in LMFDB)
4.5.b_e_ai_ap	$3$	(not in LMFDB)