Properties

Label 1.173.v
Base field $\F_{173}$
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:  $\F_{173}$
Dimension:  $1$
L-polynomial:  $1 + 21 x + 173 x^{2}$
Frobenius angles:  $\pm0.794267168102$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-251}) \)
Galois group:  $C_2$
Jacobians:  $7$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $195$ $29835$ $5176080$ $895795875$ $154963107975$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $195$ $29835$ $5176080$ $895795875$ $154963107975$ $26808761004480$ $4637914300983795$ $802359177683587875$ $138808137897412336080$ $24013807852306033098675$

Jacobians and polarizations

This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{173}$.

Endomorphism algebra over $\F_{173}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-251}) \).

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
1.173.av$2$(not in LMFDB)