Every fingering that fits a hand
3131
moderate · no 5
3534
moderate · barre
313x
easy · no 5
3134
hard · no 5
xa9a
easy · no 5
What is true of it
A♯m6 is A♯, C♯, E♯ and F𝄪 — 1, b3, 5 and 6 counted from A♯.
F𝄪 is a double sharp, and it has to be: every letter from A♯ upwards is used once, so the 6 cannot be written any other way.
These are also the notes of Gm7♭5/A♯. Which name is right depends on what is underneath and what came before — the notes alone do not decide.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 5 fingerings below is 3131: 4 fingers, held at the 1st fret.
None of them uses an open string, so this chord moves up the neck unchanged — the same shape 1 fret higher is the chord a semitone above.
1 of the 5 need a barre.
It leaves out the 5. 4 strings cannot carry 4 notes, and the 5 is the one this chord can lose without changing what it is.
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♯m6/C♯
x131
A♯m6/E♯
aa9a
A♯m6/F𝄪
0111
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.