# Properties

 Label 1.389.abi Base Field $\F_{389}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{389}$ Dimension: $1$ L-polynomial: $1 - 34 x + 389 x^{2}$ Frobenius angles: $\pm0.169253027330$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-1})$$ 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 $A(\F_{q^r})$ 356 150944 58864244 22898204800 8907344807236 3464955191237024 1347867525591145684 524320466719938163200 203960661546103572116516 79340697341449830669228704

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 356 150944 58864244 22898204800 8907344807236 3464955191237024 1347867525591145684 524320466719938163200 203960661546103572116516 79340697341449830669228704

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{389}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-1})$$.
All geometric endomorphisms are defined over $\F_{389}$.

## 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 1.389.bi $2$ (not in LMFDB) 1.389.au $4$ (not in LMFDB) 1.389.u $4$ (not in LMFDB)