# Properties

 Label 1.397.abi Base Field $\F_{397}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{397}$ Dimension: $1$ L-polynomial: $1 - 34 x + 397 x^{2}$ Frobenius angles: $\pm0.174655262871$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 14

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 14 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 364 157248 62571964 24840781056 9861722751244 3915101757542976 1554295350533798236 617055253420108059648 244970935601304037979308 97253461433791920286306368

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 364 157248 62571964 24840781056 9861722751244 3915101757542976 1554295350533798236 617055253420108059648 244970935601304037979308 97253461433791920286306368

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{397}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{397}$.

## 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 1.397.bi $2$ (not in LMFDB) 1.397.ab $3$ (not in LMFDB) 1.397.bj $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.397.bi $2$ (not in LMFDB) 1.397.ab $3$ (not in LMFDB) 1.397.bj $3$ (not in LMFDB) 1.397.abj $6$ (not in LMFDB) 1.397.b $6$ (not in LMFDB)