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

• Label: 4.5.al_cc_agj_pe
• 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:	$( 1 - 4 x + 5 x^{2} )^{2}( 1 - 3 x + 4 x^{2} - 15 x^{3} + 25 x^{4} )$
	$1 - 11 x + 54 x^{2} - 165 x^{3} + 394 x^{4} - 825 x^{5} + 1350 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0673911931187$, $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.599275473548$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$48$	$230400$	$173606976$	$148163788800$	$101953185785328$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$13$	$82$	$605$	$3335$	$15946$	$78731$	$393885$	$1957834$	$9763573$

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 $\F_{5^{3}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae^2 $\times$ 2.5.ad_e 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_e : \(\Q(\sqrt{-3}, \sqrt{-11})\).

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

The base change of $A$ to $\F_{5^{3}}$ is 1.125.as^2 $\times$ 1.125.ae^2. The endomorphism algebra for each factor is:

• 1.125.as^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
• 1.125.ae^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.af_g_v_adi	$2$	(not in LMFDB)
4.5.ad_ac_d_ba	$2$	(not in LMFDB)
4.5.d_ac_ad_ba	$2$	(not in LMFDB)
4.5.f_g_av_adi	$2$	(not in LMFDB)
4.5.l_cc_gj_pe	$2$	(not in LMFDB)
4.5.ac_ad_ag_cm	$3$	(not in LMFDB)
4.5.b_d_am_aba	$3$	(not in LMFDB)
4.5.k_cc_hk_tf	$3$	(not in LMFDB)