Properties

Label 4.5.al_cj_aip_wm
Base Field $\F_{5}$
Dimension $4$
$p$-rank $4$
Does not contain a Jacobian

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Invariants

Base field:  $\F_{5}$
Dimension:  $4$
Weil polynomial:  $( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4} )$
Frobenius angles:  $\pm0.0878807261908$, $\pm0.14758361765$, $\pm0.35241638235$, $\pm0.450170915301$
Angle rank:  $3$ (numerical)

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $4$
Slopes:  $[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 72 397440 272519208 144731750400 92214761542272 60251969345892480 37826217127905395688 23430229459787612160000 14576431850763169422818952 9100385940366051423171379200

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

Decomposition

1.5.ae $\times$ 1.5.ac $\times$ 2.5.af_n

Base change

This is a primitive isogeny class.