Properties

Label 1.449.abq
Base field $\F_{449}$
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:  $\F_{449}$
Dimension:  $1$
L-polynomial:  $1 - 42 x + 449 x^{2}$
Frobenius angles:  $\pm0.0426157488015$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-2}) \)
Galois group:  $C_2$
Jacobians:  $3$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $408$ $200736$ $90501336$ $40642616448$ $18248683777368$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $408$ $200736$ $90501336$ $40642616448$ $18248683777368$ $8193661898582304$ $3678954246632617752$ $1651850457718887756288$ $741680855532654028285464$ $333014704134429944348748576$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{449}$.

Endomorphism algebra over $\F_{449}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}) \).

Base change

This is a primitive isogeny class.

Twists

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

TwistExtension degreeCommon base change
1.449.bq$2$(not in LMFDB)