Properties

Label 2.193.abw_bkz
Base Field $\F_{193}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{193}$
Dimension:  $2$
L-polynomial:  $( 1 - 25 x + 193 x^{2} )( 1 - 23 x + 193 x^{2} )$
Frobenius angles:  $\pm0.143734387197$, $\pm0.189598946136$
Angle rank:  $1$ (numerical)
Jacobians:  31

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 31 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 28899 1373367177 51682553604864 1925222059351192329 71709354098056097496099 2671086347119640841444458496 99495248097890118081285007086627 3706098388334656603753467367532975625 138048458700232695871510244219046599099136 5142167038096105580768855004541634568892330377

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 146 36868 7189058 1387559428 267786861746 51682566660478 9974730628906994 1925122955245673476 371548729913362368194 71708904872876447192068

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{193}$
The isogeny class factors as 1.193.az $\times$ 1.193.ax 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 $\overline{\F}_{193}$
The base change of $A$ to $\F_{193^{6}}$ is 1.51682540549249.bcovba 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
All geometric endomorphisms are defined over $\F_{193^{6}}$.
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
2.193.ac_ahh$2$(not in LMFDB)
2.193.c_ahh$2$(not in LMFDB)
2.193.bw_bkz$2$(not in LMFDB)
2.193.abb_qu$3$(not in LMFDB)
2.193.av_nc$3$(not in LMFDB)
2.193.a_ajf$3$(not in LMFDB)
2.193.a_afn$3$(not in LMFDB)
2.193.a_os$3$(not in LMFDB)
2.193.v_nc$3$(not in LMFDB)
2.193.bb_qu$3$(not in LMFDB)
2.193.bw_bkz$3$(not in LMFDB)
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.193.ac_ahh$2$(not in LMFDB)
2.193.c_ahh$2$(not in LMFDB)
2.193.bw_bkz$2$(not in LMFDB)
2.193.abb_qu$3$(not in LMFDB)
2.193.av_nc$3$(not in LMFDB)
2.193.a_ajf$3$(not in LMFDB)
2.193.a_afn$3$(not in LMFDB)
2.193.a_os$3$(not in LMFDB)
2.193.v_nc$3$(not in LMFDB)
2.193.bb_qu$3$(not in LMFDB)
2.193.bw_bkz$3$(not in LMFDB)
2.193.aby_bmx$6$(not in LMFDB)
2.193.abu_bjf$6$(not in LMFDB)
2.193.az_qq$6$(not in LMFDB)
2.193.ax_my$6$(not in LMFDB)
2.193.ae_pa$6$(not in LMFDB)
2.193.e_pa$6$(not in LMFDB)
2.193.x_my$6$(not in LMFDB)
2.193.z_qq$6$(not in LMFDB)
2.193.bu_bjf$6$(not in LMFDB)
2.193.by_bmx$6$(not in LMFDB)
2.193.a_aos$12$(not in LMFDB)
2.193.a_fn$12$(not in LMFDB)
2.193.a_jf$12$(not in LMFDB)