From output s37check_all.log3 we find for the equivalence classes:

1 of size 13
0 of size 12
0 of size 11
0 of size 10
2 of size 9
5 of size 8
2 of size 7
4 of size 6
7 of size 5
12 of size 4
28 of size 3
103 of size 2
325 of size 1

making a total of 489 equivalence classes.

Check: 
1*325+2*103+3*28+4*12+5*7+6*4+7*2+8*5+9*2+13*1 = 807
is the total number of matrices, and
1*325+3*103+6*28+10*12+15*7+21*4+28*2+36*5+45*2+91*1 = 1528
is the total number of pairs considered.

There are 284 different values of the determinant.  
Most popular are

d = 1200757082375992968 = 648*9^16 (31 times)
d = 1204463122753696650 = 650*9^16 (30 times)

(these d values are corrected for rounding errors; 
the first value is the lower bound).