# Properties

 Label 1.397.k 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: $1 + 10 x + 397 x^{2}$ Frobenius angles: $\pm0.580740609043$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-93})$$ Galois group: $C_2$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 408 158304 62559864 24840430464 9861722957208 3915101639930976 1554295346184230904 617055253426333171200 244970935602275075309208 97253461433789493013748064

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 408 158304 62559864 24840430464 9861722957208 3915101639930976 1554295346184230904 617055253426333171200 244970935602275075309208 97253461433789493013748064

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{397}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-93})$$.
All geometric endomorphisms are defined over $\F_{397}$.

## 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 1.397.ak $2$ (not in LMFDB)