I'm tuned a wee bit lower than double C, more like double B flat or even double A! But everything is relative......
So here are the chords I mentioned. The numbers show the fret positions from 5th to 1st strings. An X means don't sound that string.. ..

C major 0,0,0,0,2 or 0,0,5,4,5 (root)
D minor 0,2,2,2,3
E minor 0,4,4,4,5
F major 0,0,2,0,3 or 0,0,5,5,7 or 0,0,10,9,10 (4th)
G major 0,2,0,2,0 or 0,7,7,7,9 or 0,X,12,11,12 (5th)
A minor 0,0,2,0,2 (relative minor)
B minor x,2,4,2,4

And though not part of a C major scale, here are some other useful chords:
C7 0,0,3,0,2
F7 0,0,2,3,3
G7 0,2,0,2,3

and also not in the scale of C major, and because rules are for breaking, a B flat is quite common
B Flat 0,0,3,2,3

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