Properties

Label 1.449.ai
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

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Invariants

Base field:  $\F_{449}$
Dimension:  $1$
L-polynomial:  $1 - 8 x + 449 x^{2}$
Frobenius angles:  $\pm0.439549394344$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-433}) \)
Galois group:  $C_2$
Jacobians:  $12$

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $442$ $202436$ $90529114$ $40642670848$ $18248683529882$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $442$ $202436$ $90529114$ $40642670848$ $18248683529882$ $8193662099972804$ $3678954252628747898$ $1651850457753655231488$ $741680855531564287595386$ $333014704134426566845435076$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 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 \(\Q(\sqrt{-433}) \).

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.i$2$(not in LMFDB)