# Properties

 Label 1.449.abq Base Field $\F_{449}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

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## 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.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $1$ Slopes: $[0, 1]$

## Point counts

This isogeny class contains the Jacobians of 3 curves, and hence is principally polarizable:

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

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

## Decomposition and endomorphism algebra

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

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.449.bq $2$ (not in LMFDB)