Every fingering that fits a hand
x111
moderate · barre · no 5
656x
easy · no 5
6564
hard · no 5
6868
moderate · barre
6568
hard · no 5
What is true of it
C♯6 is C♯, E♯, G♯ and A♯ — 1, 3, 5 and 6 counted from C♯.
These are also the notes of A♯m7/C♯ and B♭m7/C♯. 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 x111: one finger, one of them barring the 1st fret, 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.
2 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.
1 string stays silent.
Where its notes are on the neck
The brass dots are C♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
C♯6/E♯
x564
C♯6/G♯
1111
C♯6/A♯
3544
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.