# 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

## 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.

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

 $r$ 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.
 Twist Extension Degree Common 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