Every fingering that fits a hand
3210
open
3250
easy · no 5
3555
moderate
3255
hard · no 5
325x
easy · no 5
What is true of it
A♯maj7 is A♯, C𝄪, E♯ and G𝄪 — 1, 3, 5 and 7 counted from A♯.
C𝄪 and G𝄪 are double sharps, and they have to be: every letter from A♯ upwards is used once, so the 3 cannot be written any other way.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 5 fingerings below is 3210: 3 fingers, held at the 1st fret.
1 of them use open strings, which is what makes a shape quiet to change to.
Where its notes are on the neck
The brass dots are A♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
A♯maj7/C𝄪
7968
A♯maj7/E♯
aaac
A♯maj7/G𝄪
2211
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.