Properties

Label 2.17.an_cw
Base field $\F_{17}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{17}$
Dimension:  $2$
L-polynomial:  $( 1 - 8 x + 17 x^{2} )( 1 - 5 x + 17 x^{2} )$
  $1 - 13 x + 74 x^{2} - 221 x^{3} + 289 x^{4}$
Frobenius angles:  $\pm0.0779791303774$, $\pm0.292637436158$
Angle rank:  $2$ (numerical)
Jacobians:  $1$
Isomorphism classes:  3

This isogeny class is not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $130$ $77740$ $24261640$ $6990380800$ $2015239733650$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $5$ $269$ $4940$ $83697$ $1419325$ $24129506$ $410304445$ $6975740833$ $118588565420$ $2015998851389$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$
The isogeny class factors as 1.17.ai $\times$ 1.17.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.17.ad_ag$2$(not in LMFDB)
2.17.d_ag$2$(not in LMFDB)
2.17.n_cw$2$(not in LMFDB)

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

TwistExtension degreeCommon base change
2.17.ad_ag$2$(not in LMFDB)
2.17.d_ag$2$(not in LMFDB)
2.17.n_cw$2$(not in LMFDB)
2.17.ah_bs$4$(not in LMFDB)
2.17.ad_y$4$(not in LMFDB)
2.17.d_y$4$(not in LMFDB)
2.17.h_bs$4$(not in LMFDB)