Properties

Label 2.64.ax_jx
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:  $1 - 23 x + 257 x^{2} - 1472 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.187526431035$, $\pm0.292742513991$
Angle rank:  $2$ (numerical)
Number field:  4.0.2163369.2
Galois group:  $D_{4}$
Jacobians:  24

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 24 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2859 16722291 69021657792 281690353075491 1152998534250850299 4722373774647621483264 19342805126527751145407259 79228158008485812784625443779 324518552954415946125898160640192 1329227995908519922636880925620474451

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 42 4082 263295 16790050 1073813562 68719582847 4398044695002 281474960702914 18014398470401535 1152921504714056402

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.2163369.2.
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.x_jx$2$(not in LMFDB)