Properties

Label 3.4.ai_bc_acm
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:  $( 1 - 2 x )^{4}( 1 + 4 x^{2} )$
Frobenius angles:  $0.0$, $0.0$, $0.0$, $0.0$, $\pm0.5$
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
$A(\F_{q^r})$ 5 2025 156065 11390625 946609025 66556260225 4262469942785 274941996890625 17874140985554945 1150668609575450625

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -3 9 33 161 897 3969 15873 64001 260097 1046529

Decomposition

1.4.ae 2 $\times$ 1.4.a

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^2}$.

SubfieldPrimitive Model
$\F_{2}$3.2.ac_ac_i
$\F_{2}$3.2.c_ac_ai