Properties

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

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Invariants

Base field:  $\F_{2^{6}}$
Dimension:  $2$
L-polynomial:  $1 - 27 x + 309 x^{2} - 1728 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.133269407035$, $\pm0.218316916306$
Angle rank:  $2$ (numerical)
Number field:  4.0.108625.1
Galois group:  $D_{4}$
Jacobians:  12

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 12 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2651 16332811 68762316656 281615962081795 1153024108062559271 4722415865910083995456 19342829518893475864835291 79228165150401417429573640995 324518552769470505806933688128816 1329227994950016615803515468053011491

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 38 3986 262307 16785618 1073837378 68720195351 4398050241182 281474986076098 18014398460135003 1152921503882687306

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.108625.1.
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.bb_lx$2$(not in LMFDB)