Properties

Label 2.64.ax_jx
Base field $\F_{2^{6}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
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:  $24$
Isomorphism classes:  24

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2859$ $16722291$ $69021657792$ $281690353075491$ $1152998534250850299$

Point counts of the curve

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

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{6}}$.

Endomorphism algebra over $\F_{2^{6}}$
The endomorphism algebra of this simple isogeny class is 4.0.2163369.2.

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)