Properties

Label 1.307.af
Base field $\F_{307}$
Dimension $1$
$p$-rank $1$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

Related objects

Downloads

Learn more

Invariants

Base field:  $\F_{307}$
Dimension:  $1$
L-polynomial:  $1 - 5 x + 307 x^{2}$
Frobenius angles:  $\pm0.454427205872$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-1203}) \)
Galois group:  $C_2$
Jacobians:  $6$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $303$ $94839$ $28938924$ $8882715579$ $2727040150833$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $303$ $94839$ $28938924$ $8882715579$ $2727040150833$ $837202029518736$ $257021012312523699$ $78905450510309649075$ $24223973308617053114388$ $7436759805837973244107239$

Jacobians and polarizations

This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{307}$.

Endomorphism algebra over $\F_{307}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1203}) \).

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
1.307.f$2$(not in LMFDB)