# Properties

 Label 1.41.ai Base Field $\F_{41}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{41}$ Dimension: $1$ L-polynomial: $1 - 8 x + 41 x^{2}$ Frobenius angles: $\pm0.285223287477$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-1})$$ 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 $A(\F_{q^r})$ 34 1700 69394 2828800 115861154 4750019300 194753391314 7984921651200 327381941955874 13422659517342500

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 34 1700 69394 2828800 115861154 4750019300 194753391314 7984921651200 327381941955874 13422659517342500

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{41}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
All geometric endomorphisms are defined over $\F_{41}$.

## 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 1.41.i $2$ (not in LMFDB) 1.41.ak $4$ (not in LMFDB) 1.41.k $4$ (not in LMFDB)