Properties

Label 2.199.aca_bpf
Base Field $\F_{199}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{199}$
Dimension:  $2$
L-polynomial:  $1 - 52 x + 1071 x^{2} - 10348 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0588869325648$, $\pm0.170366934480$
Angle rank:  $2$ (numerical)
Number field:  4.0.803088.1
Galois group:  $D_{4}$
Jacobians:  8

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 8 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30273 1546132929 62067793800276 2459349504967905033 97393779093852815039313 3856887558317266970755007376 152736584276857284818886083566497 6048521416597917624649122380552951433 239527496518660501914580537029533727200532 9485528389624831234402903864610759960558223969

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 148 39040 7876024 1568223460 312079926988 62103849313174 12358664401491556 2459374192577916868 489415464118611355624 97393677359494378528480

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.803088.1.
All geometric endomorphisms are defined over $\F_{199}$.

Base change

This is a primitive isogeny class.

Twists

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