Sharedwww / Tables / degphi_r_analysis.txtOpen in CoCalc
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> ]