Properties

Label 4.5.al_cj_ain_wd
Base Field $\F_{5}$
Dimension $4$
$p$-rank $3$
Does not contain a Jacobian

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Invariants

Base field:  $\F_{5}$
Dimension:  $4$
Weil polynomial:  $( 1 - 3 x + 5 x^{2} )( 1 - 8 x + 32 x^{2} - 85 x^{3} + 160 x^{4} - 200 x^{5} + 125 x^{6} )$
Frobenius angles:  $\pm0.0657033817182$, $\pm0.238557099512$, $\pm0.265942140215$, $\pm0.475140873389$
Angle rank:  $4$ (numerical)

Newton polygon

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

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
$A(\F_{q^r})$ 75 412425 285123600 156154415625 96915522862875 60075452420942400 36982220554645153275 23089333365449331215625 14518910772898452925516800 9102589711207377594106379625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -5 27 145 643 3175 15747 77555 387363 1948690 9773827

Decomposition

1.5.ad $\times$ 3.5.ai_bg_adh

Base change

This is a primitive isogeny class.