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

• Label: 4.5.al_ci_aih_vm
• 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:	$( 1 - 4 x + 5 x^{2} )( 1 - 7 x + 27 x^{2} - 72 x^{3} + 135 x^{4} - 175 x^{5} + 125 x^{6} )$
	$1 - 11 x + 60 x^{2} - 215 x^{3} + 558 x^{4} - 1075 x^{5} + 1500 x^{6} - 1375 x^{7} + 625 x^{8}$
Frobenius angles:	$\pm0.0767028971338$, $\pm0.147583617650$, $\pm0.313688588913$, $\pm0.486729365003$
Angle rank:	$4$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$68$	$368560$	$253973744$	$145725675520$	$95113205467828$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$25$	$130$	$597$	$3115$	$15838$	$78451$	$390973$	$1957486$	$9782145$

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 $\times$ 3.5.ah_bb_acu 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 : \(\Q(\sqrt{-1}) \).
• 3.5.ah_bb_acu : 6.0.31554496.1.

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.ad_e_b_as	$2$	(not in LMFDB)
4.5.d_e_ab_as	$2$	(not in LMFDB)
4.5.l_ci_ih_vm	$2$	(not in LMFDB)