Properties

Label 3.4.ah_z_ach
Base Field $\F_{2^{2}}$
Dimension $3$
Ordinary Yes
$p$-rank $3$
Principally polarizable Yes
Contains a Jacobian No

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $3$
L-polynomial:  $( 1 - 3 x + 4 x^{2} )( 1 - 4 x + 9 x^{2} - 16 x^{3} + 16 x^{4} )$
Frobenius angles:  $\pm0.117169895439$, $\pm0.230053456163$, $\pm0.478661301576$
Angle rank:  $3$ (numerical)

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 12 4416 291708 16533504 1131951612 72889661376 4464561305244 279809450459136 17975100366192348 1155299571951176256

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -2 18 70 254 1078 4338 16630 65150 261574 1050738

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The isogeny class factors as 1.4.ad $\times$ 2.4.ae_j and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{2}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
3.4.ab_b_af$2$3.16.b_ab_db
3.4.b_b_f$2$3.16.b_ab_db
3.4.h_z_ch$2$3.16.b_ab_db