# Properties

 Label 1.337.abi Base Field $\F_{337}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{337}$ Dimension: $1$ L-polynomial: $1 - 34 x + 337 x^{2}$ Frobenius angles: $\pm0.123182934465$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 304 113088 38267824 12897912576 4346599770544 1464803674439616 493638821956772656 166356282595288194048 56062067226374219161648 18892916655144219773842368

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 304 113088 38267824 12897912576 4346599770544 1464803674439616 493638821956772656 166356282595288194048 56062067226374219161648 18892916655144219773842368

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{337}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{337}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.337.bi $2$ (not in LMFDB) 1.337.f $3$ (not in LMFDB) 1.337.bd $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.337.bi $2$ (not in LMFDB) 1.337.f $3$ (not in LMFDB) 1.337.bd $3$ (not in LMFDB) 1.337.abd $6$ (not in LMFDB) 1.337.af $6$ (not in LMFDB)