Properties

Label 2.17.ak_cf
Base field $\F_{17}$
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:  $\F_{17}$
Dimension:  $2$
L-polynomial:  $1 - 10 x + 57 x^{2} - 170 x^{3} + 289 x^{4}$
Frobenius angles:  $\pm0.216316574334$, $\pm0.356804757520$
Angle rank:  $2$ (numerical)
Number field:  4.0.94784.1
Galois group:  $D_{4}$
Jacobians:  $3$
Isomorphism classes:  3

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $167$ $88009$ $25138844$ $7030951001$ $2016631399207$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $8$ $304$ $5114$ $84180$ $1420308$ $24134518$ $410336564$ $6975796644$ $118587739898$ $2015990823264$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17}$.

Endomorphism algebra over $\F_{17}$
The endomorphism algebra of this simple isogeny class is 4.0.94784.1.

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
2.17.k_cf$2$(not in LMFDB)