Properties

Label 2.199.abx_bmc
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:  $1 - 49 x + 990 x^{2} - 9751 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0779236108176$, $\pm0.221959774337$
Angle rank:  $2$ (numerical)
Number field:  4.0.4178493.1
Galois group:  $D_{4}$
Jacobians:  48

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 48 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 30792 1551670464 62093032528608 2459421500478281472 97393886996583185842872 3856887411085056936212388864 152736582694639485750120920461656 6048521411117119397211690496995621888 239527496510031697415148796759176901979616 9485528389644232513285062059761005443514504384

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 151 39181 7879228 1568269369 312080272741 62103846942430 12358664273466571 2459374190349383185 489415464100980518164 97393677359693583234661

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.4178493.1.
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.bx_bmc$2$(not in LMFDB)