Properties

Label 2.8.ae_j
Base field $\F_{2^{3}}$
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:  $\F_{2^{3}}$
Dimension:  $2$
L-polynomial:  $1 - 4 x + 9 x^{2} - 32 x^{3} + 64 x^{4}$
Frobenius angles:  $\pm0.110959553988$, $\pm0.574771923910$
Angle rank:  $2$ (numerical)
Number field:  4.0.218768.2
Galois group:  $D_{4}$
Jacobians:  $6$
Isomorphism classes:  6

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $38$ $4180$ $237158$ $16385600$ $1086804598$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $67$ $461$ $3999$ $33165$ $262819$ $2096477$ $16787391$ $134260733$ $1073757507$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{3}}$.

Endomorphism algebra over $\F_{2^{3}}$
The endomorphism algebra of this simple isogeny class is 4.0.218768.2.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.8.e_j$2$2.64.c_abv