Properties

Label 2.17.ak_by
Base Field $\F_{17}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{17}$
Dimension:  $2$
L-polynomial:  $( 1 - 8 x + 17 x^{2} )( 1 - 2 x + 17 x^{2} )$
Frobenius angles:  $\pm0.0779791303774$, $\pm0.422020869623$
Angle rank:  $1$ (numerical)
Jacobians:  10

This isogeny class is not 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 10 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 160 83200 24088480 6922240000 2011667984800 582622284524800 168396666261788320 48662075833712640000 14063067032705334535840 4064231406646822197280000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 8 290 4904 82878 1416808 24137570 410384584 6975884158 118587729608 2015993900450

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{17}$
The isogeny class factors as 1.17.ai $\times$ 1.17.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{17}$
The base change of $A$ to $\F_{17^{4}}$ is 1.83521.amk 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
All geometric endomorphisms are defined over $\F_{17^{4}}$.
Remainder of endomorphism lattice by field

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.ag_s$2$(not in LMFDB)
2.17.g_s$2$(not in LMFDB)
2.17.k_by$2$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.17.ag_s$2$(not in LMFDB)
2.17.g_s$2$(not in LMFDB)
2.17.k_by$2$(not in LMFDB)
2.17.aq_du$4$(not in LMFDB)
2.17.ae_bm$4$(not in LMFDB)
2.17.a_abe$4$(not in LMFDB)
2.17.a_be$4$(not in LMFDB)
2.17.e_bm$4$(not in LMFDB)
2.17.q_du$4$(not in LMFDB)
2.17.a_aq$8$(not in LMFDB)
2.17.a_q$8$(not in LMFDB)
2.17.ai_bv$12$(not in LMFDB)
2.17.ac_an$12$(not in LMFDB)
2.17.c_an$12$(not in LMFDB)
2.17.i_bv$12$(not in LMFDB)