Properties

Label 2.64.aw_iv
Base Field $\F_{2^{6}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

Learn more about

Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $1 - 22 x + 229 x^{2} - 1408 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.0819912419852$, $\pm0.366229297515$
Angle rank:  $2$ (numerical)
Number field:  4.0.88625.1
Galois group:  $D_{4}$
Jacobians:  60

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 60 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2896 16669376 68782725136 281419406769920 1152841168277570896 4722336250153830286016 19342821021891933521998096 79228176174304021085513784320 324518559836117046997926595379536 1329227996666643093343179711184871616

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 43 4071 262387 16773903 1073667003 68719036791 4398048309187 281475025240863 18014398852412683 1152921505371623431

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.88625.1.
All geometric endomorphisms are defined over $\F_{2^{6}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.64.w_iv$2$(not in LMFDB)