Properties

Label 2.193.abz_bnt
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:  $1 - 51 x + 1033 x^{2} - 9843 x^{3} + 37249 x^{4}$
Frobenius angles:  $\pm0.0593833791648$, $\pm0.174852758199$
Angle rank:  $2$ (numerical)
Number field:  4.0.11661.1
Galois group:  $D_{4}$
Jacobians:  18

This isogeny class is simple and geometrically 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 18 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 28389 1367696853 51652855050525 1925107659024551109 71708995610628624309504 2671085407704292377641103525 99495246074139766125809151266541 3706098385107069199047505841397459909 138048458698494131157967690198488030675525 5142167038109095150214457818647888207987888128

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 143 36715 7184927 1387476979 267785523038 51682548483835 9974730426019271 1925122953569111779 371548729908683130791 71708904873057590237950

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{193}$
The endomorphism algebra of this simple isogeny class is 4.0.11661.1.
All geometric endomorphisms are defined over $\F_{193}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.193.bz_bnt$2$(not in LMFDB)