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> ]