Properties

Label 2.64.ay_jz
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:  $1 - 24 x + 259 x^{2} - 1536 x^{3} + 4096 x^{4}$
Frobenius angles:  $\pm0.0708266491475$, $\pm0.324195059058$
Angle rank:  $2$ (numerical)
Number field:  4.0.390897.1
Galois group:  $D_{4}$
Jacobians:  48

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 48 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 2796 16541136 68775803892 281470388136768 1152864272724944316 4722322572286774424784 19342800643058989688984388 79228167212662731011436717312 324518560357295753462508540084108 1329227999086142731728681192482661456

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 41 4039 262361 16776943 1073688521 68718837751 4398043675577 281474993402719 18014398881343913 1152921507470204839

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.390897.1.
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.y_jz$2$(not in LMFDB)