Properties

Label 1.193.an
Base field $\F_{193}$
Dimension $1$
$p$-rank $1$
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:  $\F_{193}$
Dimension:  $1$
L-polynomial:  $1 - 13 x + 193 x^{2}$
Frobenius angles:  $\pm0.345017855847$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-67}) \)
Galois group:  $C_2$
Jacobians:  $5$

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:  $1$
Slopes:  $[0, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $181$ $37467$ $7194388$ $1387515411$ $267784511821$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $181$ $37467$ $7194388$ $1387515411$ $267784511821$ $51682526518464$ $9974730273372829$ $1925122954942698723$ $371548729949828783764$ $71708904873362421978507$

Jacobians and polarizations

This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{193}$.

Endomorphism algebra over $\F_{193}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-67}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
1.193.n$2$(not in LMFDB)