# Properties

 Label 1.137.w Base Field $\F_{137}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{137}$ Dimension: $1$ L-polynomial: $1 + 22 x + 137 x^{2}$ Frobenius angles: $\pm0.888982741545$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-1})$$ Galois group: $C_2$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 160 18560 2572960 352268800 48261648800 6611858814080 905824260302240 124097930629171200 17001416397325678240 2329194047654191260800

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 160 18560 2572960 352268800 48261648800 6611858814080 905824260302240 124097930629171200 17001416397325678240 2329194047654191260800

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{137}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
All geometric endomorphisms are defined over $\F_{137}$.

## 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 1.137.aw $2$ (not in LMFDB) 1.137.ai $4$ (not in LMFDB) 1.137.i $4$ (not in LMFDB)