The following MAGMA program was used to create the table below. Here's the email I wrote to Amod Agashe: Amod, Here's a table that gives just the levels at which there is a defect between deg(phi) and r. Along with the level, it gives the factorization of GCD((r/deg(phi))^oo, N), i.e., the powers of the prime p dividing N at which r doesn't equal deg(phi). There are no examples in which r=/=deg(phi) at primes whose square doesn't divide the level. -- William Amod's response: Your table looks very interesting. You have primes as high as 13 dividing r/deg(phi) when their squares divide N. I think for higher dimensions, your earlier calculations even had primes as high as 19 doing the same. > There are no examples in which r=/=deg(phi) at primes whose > square doesn't divide the level. OK, that makes sense. At the moment, the only thing I can claim is that deg(phi) divides r and if a prime p divides r/deg(phi) then p^2 | N or p| gcd(deg(phi),N) (for higher dimensions, you have the corresponding statements for the annihilators). So long as your data does not violate the above, I can't see anything wrong with it. Your data seem to suggest that the second reason above is not sufficient, i.e., if a prime p divides r/deg(phi) then p^2 | N. --Amod ////////////////////////////////////////////////// load "degphi_r_table.m"; procedure test1(N,d) for E in d do if E[2] ne E[3] then dif := E[3] div E[2]; N,"\t: ", [f : f in Factorization(N) | GCD(f[1],dif) gt 1]; if N mod dif ne 0 then "WARNING, N = ", N; end if; end if; end for; end procedure; procedure iterate() for N in [1..#dat] do if not (dat[N] cmpeq []) then test1(N,dat[N]); end if; end for; end procedure; ////////////////////////////////////////////////// 54 : [ <3, 3> ] 64 : [ <2, 6> ] 72 : [ <2, 3> ] 80 : [ <2, 4> ] 88 : [ <2, 3> ] 92 : [ <2, 2> ] 96 : [ <2, 5> ] 96 : [ <2, 5> ] 99 : [ <3, 2> ] 108 : [ <3, 3> ] 112 : [ <2, 4> ] 112 : [ <2, 4> ] 112 : [ <2, 4> ] 120 : [ <2, 3> ] 124 : [ <2, 2> ] 126 : [ <3, 2> ] 128 : [ <2, 7> ] 128 : [ <2, 7> ] 128 : [ <2, 7> ] 128 : [ <2, 7> ] 135 : [ <3, 3> ] 144 : [ <2, 4> ] 144 : [ <2, 4> ] 147 : [ <7, 2> ] 150 : [ <5, 2> ] 152 : [ <2, 3> ] 153 : [ <3, 2> ] 153 : [ <3, 2> ] 160 : [ <2, 5> ] 160 : [ <2, 5> ] 162 : [ <3, 4> ] 162 : [ <3, 4> ] 162 : [ <3, 4> ] 162 : [ <3, 4> ] 168 : [ <2, 3> ] 168 : [ <2, 3> ] 171 : [ <3, 2> ] 175 : [ <5, 2> ] 176 : [ <2, 4> ] 176 : [ <2, 4> ] 176 : [ <2, 4> ] 184 : [ <2, 3> ] 184 : [ <2, 3> ] 184 : [ <2, 3> ] 189 : [ <3, 3> ] 189 : [ <3, 3> ] 189 : [ <3, 3> ] 192 : [ <2, 6> ] 192 : [ <2, 6> ] 192 : [ <2, 6> ] 192 : [ <2, 6> ] 196 : [ <7, 2> ] 200 : [ <2, 3> ] 200 : [ <2, 3> ] 200 : [ <5, 2> ] 200 : [ <2, 3>, <5, 2> ] 208 : [ <2, 4> ] 208 : [ <2, 4> ] 208 : [ <2, 4> ] 216 : [ <2, 3>, <3, 3> ] 216 : [ <3, 3> ] 216 : [ <2, 3> ] 224 : [ <2, 5> ] 224 : [ <2, 5> ] 225 : [ <5, 2> ] 234 : [ <3, 2> ] 234 : [ <3, 2> ] 234 : [ <3, 2> ] 236 : [ <2, 2> ] 240 : [ <2, 4> ] 240 : [ <2, 4> ] 240 : [ <2, 4> ] 240 : [ <2, 4> ] 242 : [ <11, 2> ] 243 : [ <3, 5> ] 243 : [ <3, 5> ] 245 : [ <7, 2> ] 248 : [ <2, 3> ] 248 : [ <2, 3> ] 256 : [ <2, 8> ] 256 : [ <2, 8> ] 256 : [ <2, 8> ] 256 : [ <2, 8> ] 260 : [ <2, 2> ] 264 : [ <2, 3> ] 270 : [ <3, 3> ] 270 : [ <3, 3> ] 270 : [ <3, 3> ] 272 : [ <2, 4> ] 272 : [ <2, 4> ] 272 : [ <2, 4> ] 272 : [ <2, 4> ] 275 : [ <5, 2> ] 280 : [ <2, 3> ] 280 : [ <2, 3> ] 288 : [ <2, 5>, <3, 2> ] 288 : [ <2, 5>, <3, 2> ] 288 : [ <2, 5> ] 288 : [ <2, 5> ] 288 : [ <2, 5> ] 294 : [ <7, 2> ] 294 : [ <7, 2> ] 294 : [ <7, 2> ] 296 : [ <2, 3> ] 296 : [ <2, 3> ] 297 : [ <3, 3> ] 297 : [ <3, 3> ] 300 : [ <5, 2> ] 300 : [ <5, 2> ] 304 : [ <2, 4> ] 304 : [ <2, 4> ] 304 : [ <2, 4> ] 304 : [ <2, 4> ] 304 : [ <2, 4> ] 304 : [ <2, 4> ] 312 : [ <2, 3> ] 312 : [ <2, 3> ] 312 : [ <2, 3> ] 312 : [ <2, 3> ] 312 : [ <2, 3> ] 312 : [ <2, 3> ] 315 : [ <3, 2> ] 320 : [ <2, 6> ] 320 : [ <2, 6> ] 320 : [ <2, 6> ] 320 : [ <2, 6> ] 320 : [ <2, 6> ] 320 : [ <2, 6> ] 324 : [ <3, 4> ] 324 : [ <3, 4> ] 324 : [ <3, 4> ] 324 : [ <3, 4> ] 325 : [ <5, 2> ] 333 : [ <3, 2> ] 333 : [ <3, 2> ] 333 : [ <3, 2> ] 336 : [ <2, 4> ] 336 : [ <2, 4> ] 336 : [ <2, 4> ] 336 : [ <2, 4> ] 336 : [ <2, 4> ] 336 : [ <2, 4> ] 338 : [ <13, 2> ] 338 : [ <13, 2> ] 340 : [ <2, 2> ] 342 : [ <3, 2> ] 342 : [ <3, 2> ] 344 : [ <2, 3> ] 348 : [ <2, 2> ] 348 : [ <2, 2> ] 350 : [ <5, 2> ] 350 : [ <5, 2> ] 350 : [ <5, 2> ] 352 : [ <2, 5> ] 352 : [ <2, 5> ] 352 : [ <2, 5> ] 352 : [ <2, 5> ] 352 : [ <2, 5> ] 352 : [ <2, 5> ] 360 : [ <2, 3> ] 360 : [ <3, 2> ] 360 : [ <2, 3>, <3, 2> ] 363 : [ <11, 2> ] 368 : [ <2, 4> ] 368 : [ <2, 4> ] 368 : [ <2, 4> ] 368 : [ <2, 4> ] 368 : [ <2, 4> ] 368 : [ <2, 4> ] 368 : [ <2, 4> ] 378 : [ <3, 3> ] 378 : [ <3, 3> ] 378 : [ <3, 3> ] 378 : [ <3, 3> ] 378 : [ <3, 3> ] 378 : [ <3, 3> ] 384 : [ <2, 7> ] 384 : [ <2, 7> ] 384 : [ <2, 7> ] 384 : [ <2, 7> ] 384 : [ <2, 7> ] 384 : [ <2, 7> ] 384 : [ <2, 7> ] 384 : [ <2, 7> ] 387 : [ <3, 2> ] 392 : [ <2, 3>, <7, 2> ] 392 : [ <2, 3> ] 392 : [ <2, 3> ] 392 : [ <2, 3>, <7, 2> ] 400 : [ <2, 4> ] 400 : [ <2, 4>, <5, 2> ] 400 : [ <2, 4>, <5, 2> ] 400 : [ <2, 4> ] 400 : [ <2, 4> ] 400 : [ <2, 4> ] 400 : [ <2, 4> ] 400 : [ <2, 4> ] 405 : [ <3, 4> ] 405 : [ <3, 4> ] 405 : [ <3, 4> ] 405 : [ <3, 4> ] 405 : [ <3, 4> ] 405 : [ <3, 4> ] 408 : [ <2, 3> ] 408 : [ <2, 3> ] 414 : [ <3, 2> ] 416 : [ <2, 5> ] 416 : [ <2, 5> ] 423 : [ <3, 2> ] 425 : [ <5, 2> ] 425 : [ <5, 2> ] 428 : [ <2, 2> ] 432 : [ <2, 4>, <3, 3> ] 432 : [ <2, 4> ] 432 : [ <2, 4> ] 432 : [ <2, 4>, <3, 3> ] 432 : [ <2, 4>, <3, 3> ] 432 : [ <2, 4>, <3, 3> ] 432 : [ <2, 4>, <3, 3> ] 432 : [ <2, 4>, <3, 3> ] 440 : [ <2, 3> ] 440 : [ <2, 3> ] 440 : [ <2, 3> ] 441 : [ <7, 2> ] 441 : [ <3, 2> ] 441 : [ <7, 2> ] 448 : [ <2, 6> ] 448 : [ <2, 6> ] 448 : [ <2, 6> ] 448 : [ <2, 6> ] 448 : [ <2, 6> ] 448 : [ <2, 6> ] 448 : [ <2, 6> ] 448 : [ <2, 6> ] 450 : [ <5, 2> ] 456 : [ <2, 3> ] 456 : [ <2, 3> ] 459 : [ <3, 3> ] 459 : [ <3, 3> ] 459 : [ <3, 3> ] 459 : [ <3, 3> ] 464 : [ <2, 4> ] 464 : [ <2, 4> ] 464 : [ <2, 4> ] 464 : [ <2, 4> ] 464 : [ <2, 4> ] 464 : [ <2, 4> ] 468 : [ <3, 2> ] 468 : [ <3, 2> ] 472 : [ <2, 3> ] 475 : [ <5, 2> ] 475 : [ <5, 2> ] 477 : [ <3, 2> ] 480 : [ <2, 5> ] 480 : [ <2, 5> ] 480 : [ <2, 5> ] 480 : [ <2, 5> ] 480 : [ <2, 5> ] 480 : [ <2, 5> ] 480 : [ <2, 5> ] 480 : [ <2, 5> ] 486 : [ <3, 5> ] 486 : [ <3, 5> ] 486 : [ <3, 5> ] 486 : [ <3, 5> ] 486 : [ <3, 5> ] 486 : [ <3, 5> ] 490 : [ <7, 2> ] 490 : [ <7, 2> ] 490 : [ <7, 2> ] 490 : [ <7, 2> ] 490 : [ <7, 2> ] 495 : [ <3, 2> ] 496 : [ <2, 4> ] 496 : [ <2, 4> ] 496 : [ <2, 4> ] 496 : [ <2, 4> ] 496 : [ <2, 4> ] 496 : [ <2, 4> ] 504 : [ <2, 3>, <3, 2> ] 504 : [ <2, 3> ] 504 : [ <3, 2> ] 504 : [ <2, 3> ] 504 : [ <2, 3>, <3, 2> ] 504 : [ <2, 3> ] 504 : [ <2, 3> ] 507 : [ <13, 2> ] 513 : [ <3, 3> ] 522 : [ <3, 2> ] 522 : [ <3, 2> ] 522 : [ <3, 2> ] 525 : [ <5, 2> ] 525 : [ <5, 2> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 528 : [ <2, 4> ] 539 : [ <7, 2> ]