The base term is (√2+√3+√6+√8+√16) with 4 + signs.
I found 15 conjugates each being a repetition of the base term some number of times but with a pattern of sign changes.
3 (5, 9, 12)
0101
1001
1100
4 (3, 4, 13, 15)
0011
0100
1101
1111
5 (1, 8, 12, 13, 15)
0001
1000
1100
1101
1111
6 (2, 8, 9, 10, 13, 14)
0010
1000
1001
1010
1101
1110
7 (1, 4, 5, 8, 9, 12, 13)
0001
0100
0101
1000
1001
1100
1101
7 (1, 5, 7, 10, 13, 14, 15)
0001
0101
0111
1010
1101
1110
1111
7 (2, 5, 7, 9, 11, 12, 14)
0010
0101
0111
1001
1011
1100
1110
7 (3, 5, 6, 9, 10, 12, 15)
0011
0101
0110
1001
1010
1100
1111
8 (3, 4, 6, 8, 10, 12, 13, 14)
0011
0100
0110
1000
1010
1100
1101
1110
10 (1, 3, 5, 6, 8, 9, 11, 12, 14, 15)
0001
0011
0101
0110
1000
1001
1011
1100
1110
1111
10 (1, 4, 5, 6, 8, 9, 10, 13, 14, 15)
0001
0100
0101
0110
1000
1001
1010
1101
1110
1111
11 (1, 2, 4, 5, 7, 8, 9, 11, 12, 13, 14)
0001
0010
0100
0101
0111
1000
1001
1011
1100
1101
1110
11 (1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15)
0001
0011
0100
0101
0110
1000
1001
1010
1100
1101
1111
11 (2, 3, 5, 6, 7, 9, 10, 11, 12, 14, 15)
0010
0011
0101
0110
0111
1001
1010
1011
1100
1110
1111
15 (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
patterns of sign changes in conjugate
