Properties

Label 2.199.abz_bof
Base field $\F_{199}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{199}$
Dimension:  $2$
L-polynomial:  $1 - 51 x + 1045 x^{2} - 10149 x^{3} + 39601 x^{4}$
Frobenius angles:  $\pm0.0810958877955$, $\pm0.182598387842$
Angle rank:  $2$ (numerical)
Number field:  4.0.222573.1
Galois group:  $D_{4}$
Jacobians:  $24$

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $2$
Slopes:  $[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $30447$ $1548138609$ $62078532588333$ $2459391344344332477$ $97393911071387418585072$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $149$ $39091$ $7877387$ $1568250139$ $312080349884$ $62103854990095$ $12358664468771189$ $2459374193351624035$ $489415464129052796951$ $97393677359679137558686$

Jacobians and polarizations

This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{199}$.

Endomorphism algebra over $\F_{199}$
The endomorphism algebra of this simple isogeny class is 4.0.222573.1.

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.bz_bof$2$(not in LMFDB)