Properties

Label 3.4.ai_bg_adc
Base Field $\F_{2^2}$
Dimension $3$
$p$-rank $0$
Does not contain a Jacobian

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Invariants

Base field:  $\F_{2^2}$
Dimension:  $3$
Weil polynomial:  $( 1 - 2 x )^{2}( 1 - 2 x + 4 x^{2} )^{2}$
Frobenius angles:  $0.0$, $0.0$, $\pm0.333333333333$, $\pm0.333333333333$
Angle rank:  $0$ (numerical)

Newton polygon

This isogeny class is supersingular.

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

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
$A(\F_{q^r})$ 9 3969 321489 16769025 947593089 62523502209 4262734200321 281474943156225 18084697997050881 1152921502459363329

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 17 81 257 897 3713 15873 65537 263169 1048577

Decomposition

1.4.ae $\times$ 1.4.ac 2

Base change

This is a primitive isogeny class.