Theorem B check: phi(y)=(y,y+1); want M in SL_2(Z) with M.phi = phi^p for every y in F_q minus {0,1,-1}
columns: #y = admissible y; #ok = y for which the explicit M=B D B^-1 was verified by field arithmetic;
         maxord = largest order of phi(y);  h<=5,h<=10,h<=25 = number of y admitting some M in GL_2(Z)
         with all |entries| <= 5, 10, 25 (exhaustive search; '-' = search skipped for large q)
  p   n      q     #y    #ok  maxord   h<=5  h<=10  h<=25
  2   2      4      2      2       3      2      2      2
  2   3      8      6      6       7      6      6      6
  2   4     16     14     14      15     14     14     14
  2   5     32     30     30      31      0      0     30
  2   6     64     62     62      63      8      8     56
  2   7    128    126    126     127      0      0      0
  2   8    256    254    254     255     14     14     14
  2   9    512    510    510     511      6      6      6
  2  10   1024   1022   1022    1023      2      2     32
  2  11   2048   2046   2046    2047      0      0      0
  2  12   4096   4094   4094    4095      -      -      -
  3   2      9      6      6       8      6      6      6
  3   3     27     24     24      26     15     24     24
  3   4     81     78     78      80      6     10     66
  3   5    243    240    240     242      0      0      0
  3   6    729    726    726     728     21     30     30
  3   7   2187   2184   2184    2186      0      0      0
  5   1      5      2      2       4      2      2      2
  5   2     25     22     22      24     12     22     22
  5   3    125    122    122     124      2      2     89
  5   4    625    622    622     624     12     22     30
  5   5   3125   3122   3122    3124      -      -      -
  7   1      7      4      4       6      4      4      4
  7   2     49     46     46      48     26     36     46
  7   3    343    340    340     342     10     10     13
  7   4   2401   2398   2398    2400     26     36     54
 11   1     11      8      8      10      8      8      8
 11   2    121    118    118     120     26     36     74
 11   3   1331   1328   1328    1330      8      8     14
 13   1     13     10     10      12     10     10     10
 13   2    169    166    166     168     32     56     94
 13   3   2197   2194   2194    2196     10     10     13
TOTAL admissible y checked: 21926; explicit SL_2(Z) matrix verified for 21926
root-count bound for fixed small matrices (|entries|<=3) violated: 0 times (expected 0)
Reading: for each fixed h only finitely many y in F_p-bar admit a matrix of height <= h,
so the fraction of y served by bounded matrices tends to 0 as q grows (h-columns above).
