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

• Label: 4.5.al_cj_aip_wm
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4} )$
	$1 - 11 x + 61 x^{2} - 223 x^{3} + 584 x^{4} - 1115 x^{5} + 1525 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0878807261908$, $\pm0.147583617650$, $\pm0.352416382350$, $\pm0.450170915301$
Angle rank:	$3$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$72$	$397440$	$272519208$	$144731750400$	$92214761542272$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$27$	$139$	$591$	$3020$	$15795$	$79319$	$393087$	$1956415$	$9771462$

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 $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ae $\times$ 1.5.ac $\times$ 2.5.af_n 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.ac : \(\Q(\sqrt{-1}) \).
• 2.5.af_n : 4.0.4901.1.

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

The base change of $A$ to $\F_{5^{4}}$ is 1.625.o^2 $\times$ 2.625.acl_dfd. The endomorphism algebra for each factor is:

• 1.625.o^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
• 2.625.acl_dfd : 4.0.4901.1.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{5^{2}}$
  The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.g $\times$ 2.25.b_abf. The endomorphism algebra for each factor is:

  • 1.25.ag : \(\Q(\sqrt{-1}) \).
  • 1.25.g : \(\Q(\sqrt{-1}) \).
  • 2.25.b_abf : 4.0.4901.1.

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.ah_z_act_gu	$2$	(not in LMFDB)
4.5.ad_f_b_ay	$2$	(not in LMFDB)
4.5.ab_b_h_aq	$2$	(not in LMFDB)
4.5.b_b_ah_aq	$2$	(not in LMFDB)
4.5.d_f_ab_ay	$2$	(not in LMFDB)
4.5.h_z_ct_gu	$2$	(not in LMFDB)
4.5.l_cj_ip_wm	$2$	(not in LMFDB)