Properties

Label 2.64.ax_jr
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:  $1 - 23 x + 251 x^{2} - 1472 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.136972039278$, $\pm0.322693284184$
Angle rank:  $2$ (numerical)
Number field:  4.0.1249897.1
Galois group:  $D_{4}$
Jacobians:  30

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 30 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2853 16670079 68912651556 281579588804043 1152935557991275563 4722361706912627924496 19342818730263813205467897 79228173927895616127245211603 324518561876516438943328560468732 1329227998643077522913403057943706319

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 42 4070 262881 16783450 1073754912 68719407239 4398047788134 281475017260018 18014398965677601 1152921507085907030

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.1249897.1.
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.x_jr$2$(not in LMFDB)