kummer-degrees

kummer_degree.sage (30746B)

---

 # Computes the "adelic Kummer failure", i.e. the degrees of the intersection
 # of the the Kummer extension Q(\sqrt{2^n}{G}) with the M-th cyclotomic field
 # over Q_{2^n}.
 #
 # Input: a good basis B for the torsion-free group G, organized as a list of
 # lists, and a non negative integer d. They have to satisfy the following:
 # 1. Each list B[i] contains all basis elements of 2-divisibility i.
 # 2. The basis given by B is 2-maximal, that is to say it satisfies Theorem 14
 #    of Debry-Perucca; i.e. each element of B[i] is, up to plus or minus 1, the
 #    2^i-th power of a strongly 2-indivisible rational.
 # 3. For i != d every element of B[i] is positive, and B[d][0] is the only
 #    negative element of B[d] (if d=-1, then there is no negative element).
 # 3'. Notice that the existence on negative elements in B[0] does not influence
 #    the correctness of the algorithm; in fact, the function adjust_sign
 #    produces a basis that may have negative elements of divisibility 0, and
 #    this basis is given as input for adelic_failure_gb.
 #
 # Output: a table ad_fail as described below. Call M0 the smallest positive
 # integer such that the intersection of Q(\sqrt{2^n}{G}) with Q_\infty is
 # contained in Q_M0.
 # The table ad_fail has N rows, where N is defined below.
 # Each row R=ad_fail[i] contains a variable number of pairs (d,r), where d is a
 # divisor of M0 and r is the degree of Q(\sqrt{2^{i+1}}{G}) \cap Q_d over
 # Q_{2^n}.
 # Each divisor of M appears at most once on each row, and the last element of
 # the last row is of the form (M0,r0).
 def adelic_failure_gb( B, d ):
 
     # The table to be returned (or printed at the end), as described above.
     ad_fail = []
 
     # N is such that for every n > N the adelic failure of Q(\sqrt{2^n}{G}) is
     # the same as that of Q(\sqrt{2^N}{G}).
     # We always have to include n=3, because of problem with sqrt(2) in Q_8 (in
     # theory, this is not necessary in some cases, e.g. if 2 does not divide
     # any element of G).
     # If the negative generator is on the last level, we need to increase N by
     # 1, because it would contribute to the shortlist in the next level (by
     # taking the root of an even power).
     if d == len(B)-1:
         N = max(3,len(B)+1)
     else:
         N = max(3,len(B))
 
     # The intersection is given by adding the square roots of the elements of
     # this shortlist (and a "special element", not always of the form sqrt(d),
     # coming from taking a suitable root of a negative generator; this special
     # element is dealt with later). The shortlist grows at each step, so we
     # declare it before starting to loop over n and we build it incrementally.
     # The "special element" is of the form \zeta_{2^n}\sqrt{b}, which we encode
     # as (n,b). We use the value (1,1) (special element -1) to say that there
     # isno special element at this level.
     shortlist = []
     special_element = (1,1)
 
     # The integers M, giving the smallest cyclotomic field in which lies the
     # whole intersection with Q_\infty, also grows with n. As with the
     # shortlist, we declare it here and increase it appropriately at each step.
     M = 1
 
     for n in range( 1, N+1 ): # 1 \leq n \leq N
 
         # We add the new elements to the shortlist, modifying M if needed.
         # This is not done in case we are in the extra "fake" level (this case
         # dealt with immediately below).
         if n-1 < len(B):
             for g in B[n-1]:
                 # Case of negative g
                 if g < 0 and n > 1:
                     # Special element of the form \zeta_{2^{n+1}}\sqrt(b).
                     # It is contained in Q_{lcm(2^{n+1},cyc_emb(b))}, except in 
                     # case n=2 and b=2, in which case it is contained in Q_4.
                     # We store it as as pair ( n+1, b ).
                     special_element = ( n+1, abs(g)^(1/(2^(n-1))) )
                     M = lcm( M, special_embed( special_element ) )
                 else:
                     b = g^(1/(2^(n-1))) # b is 2-indivisible
                     shortlist.append( b )
                     M = lcm( M, cyc_embed(b) )
 
         # We add a root of an even power of the negative generator, as soon as
         # we are beyond its level.
         if d != -1 and n == d+2:        
             b = abs(B[d][0])^(1/2^d)
             shortlist.append( b ) 
             M = lcm( M, cyc_embed(b) )
 
         M = lcm(M,2^n)
             
         # Here we account for the extra 2^{n+1} root of unity (in case the
         # negative element has enough divisibility).
         if n <= d:
             M = lcm( M, 2^(n+1) )
 
         # We have to add -1 to the shortlist (see example [12,36]) and
         # remove it later.
         if n == 1 and d >= 1:
             shortlist.append(-1)
         if n > 1 and -1 in shortlist:
             shortlist.remove(-1)
 
         # For each divisor dM of M, compute the degree of the intersection of
         # Q(\sqrt{2^n}{G}) with Q_dM over Q_{2^n}. We need to compute the
         # number r of elements in the subgroup of G generated by the shortlist
         # that lie in Q_dM. This is going to be a power of 2. The sought degree
         # will be r, up to considering some special cases (described below).
         # 
         # This algorithm could be inefficient for groups of big rank. It can be
         # improved by precomputing subgroups of G/G^2 and the cyclotomic fields
         # containing them.
 
         aux = [] # Next line of ad_fail table
 
         for dM in divisors( M ):
             # We only care of the intersection with Q_dM if it contains the 2^n
             # roots of unity.
             if dM % (2^n) != 0:
                 continue
 
             # The following line gives, without repetitions, the list of m's
             # such that that some element in the subgroup of G generated by
             # the shortlist embeds in Q_m.
             S = [ product(s) for s in subsets( shortlist ) ]
             H = [ cyc_embed( s ) for s in S ] 
             r = len( [ b for b in H if dM % b == 0 ] )
             
             # We double n in case a new roots of unity enters the cyclotomic
             # due to the negative negerator. In case n=1, this is accounted
             # by having -1 in the shortlist.
             if n <= d and dM % (2^(n+1)) == 0 and n > 1:
                 r *= 2
 
             # We loose a factor of 2 if we have sqrt(2) in Q_8 or \zeta_8 2 in
             # Q_4.
             if 8 in H and dM % 8 == 0 and (n >= 3 or (n == 2 and n <= d)):
                 r = r/2
 
             # If we have a special element in this level, we consider it.
             # We have to consider all possible special elements arising from
             # multiplying the given element with the other elements of the
             # shortlist.
             # If any of them is of the form \zeta_8 2q for q a square, there is
             # nothing to do: in fact \zeta_8 2 embeds in Q_4, which coincides
             # with Q_{2^n} (n must be 2); so we would double the degree because
             # of the existence of the special element, butthen we would loose
             # another factor of 2 because of this.
             if special_element != (1,1) and special_element[0] == n+1:
                 nothing_to_do = False
                 intersecting_QdM = False
                 for s in S:
                     new_special = ( n+1, special_element[1] * s )
                     m = special_embed( new_special )
                     if n == 2 and m == 4: # \zeta_8 times 2 times square
                         nothing_to_do = True
                     if dM % m == 0:
                         intersecting_QdM = True
                 if intersecting_QdM and not nothing_to_do:
                     r *= 2
 
             aux.append( (dM,r) )
         
         ad_fail.append(aux)
 
     return ad_fail
 
 # Returns the smallest m such that \sqrt(b) is in the m-th cyclotomic field.
 def cyc_embed( b ):
     m = squarefree_part(b)
     if m%4 != 1:
         m *= 4
     return abs(m)
 
 # Computes the minimal cyclotomic field containing \zeta_{2^n}\sqrt(b),
 # where (n,b)=p
 def special_embed( p ):
     n = p[0]
     b = p[1]
     m = squarefree_part(b)
     if n == 3 and m % 2 == 0:
         return 4 * cyc_embed(m/2)
     else:
         return lcm( 2^n, cyc_embed(b) )
 
 # Computes the "l-adic failure", i.e. the ratio between l^{nr} and the degree
 # of Q_{l^m}(\sqrt[2^n](G)) over Q_{l^m} for all possible l.
 # Input: any basis for G
 # Returns a pair (L,T) where:
 #   - L is a list of primes l that do not have maximal l-adic part
 #   - T is an array of tables, where T[i][m][n] is the l-valuation of the
 #     degree of Q_{l^m,l^n} over Q_{l^m}, where l = L[i]
 def total_l_adic_failure( B ):
 
     bp = bad_primes( B )
     ret = ( bp, [] )
 
     for l in bp:
         ret[1].append( l_adic_failure( B, l ) )
 
     return ret
 
 # Computes the "bad primes", i.e. the ones for which the l-adic part is not
 # maximal. Used by l_adic_degree and total_l_adic_failure.
 def bad_primes( B ):
     M = exponent_matrix( B )
     (a,b) = M.dimensions()
     if a > b or M.rank() < a:
         print( "This is not a basis" )
         return
 
     # Compute which primes l divide all minors of the exponent matrix
     ms = M.minors( a )
     d = ms[0]
     for m in ms:
         d = gcd( d, m )
     bad_primes = list( prime_factors( d ) )
     if 2 not in bad_primes:
         bad_primes += [2] # 2 is always bad
     bad_primes.sort() # Ensures 2 is always first
 
     return bad_primes
    
 # Computes the l-adic failure for a specific l. Returns a "table" as described
 # above "total_l_adic_failure".
 # B is any basis for G.
 def l_adic_failure( B, l ):
 
     r = len(B)
     GB = make_good_basis( B, l )
     
     # Computes the parameters over Q4. For l odd, they are the same as over Q.
     p = parameters_Q4( GB, l )
     maxM = max( [ sum(x) for x in p ] )
     maxN = max( maxM, len(GB)-1 )
 
     maxM = max( 1, maxM )
     if l == 2:
         maxM = max( 2, maxM ) # For computing parameters over Q4
     maxN = max( 1, maxN )
 
     tabel = []
     max_failure = 0
     for m in range( 1, maxM+1 ):
         row = []
         for n in range( 1, max( m+1, maxN ) ):
             vl = -1
             if m >= n:
                 if l==2 and n==1 and m==1:
                     vl = compute_vl( p, 1, 2, r )
                     if adjust_sign( GB )[1] >= 1:
                         vl += 1
                 else:
                     vl = compute_vl( p, n, m, r )
                 failure = r*n - vl
             row.append(vl)
             max_failure = max( max_failure, failure )
         tabel.append((row,max_failure))
     return tabel
 
 # Returns a power of l that is the l-adic failure at M, N.
 # tablel must be the output table of l_adic_failure.
 def l_adic_failure_from_data( B, l, tablel, M, N ):
 
     m = valuation( M, l )
     n = valuation( N, l )
     if m < n:
         # inconsistent choice of M and N
         return 0
     if n == 0:
         return 1
     r = len(B)
 
     # Basically, the "dual" of what we do in l_adic_degree.
     if m > len(tablel):
         if n > len(tablel):
             return l^tablel[-1][1]
         else:
             return l^(r*n-tablel[-1][0][n-1])
     
     return l^(r*n-tablel[m-1][0][n-1])
 
 # Computes the l-divisibility parameters of G over Q4, given a good basis b
 # over Q for G. Returns a list of pairs (di,hi).
 # If l is odd it just uses the good basis given to compute the parameters.
 def parameters_Q4( gb, l ):
     # Converts from "good basis format" to simple list
     b = []
     for x in gb:
         b += x
     ret = []
     
     if l != 2:
         for i in range( len( gb ) ):
             for j in gb[i]:
                 ret.append( (i,0) )
         return ret
     else:
         R.<y> = PolynomialRing( QQ )
         pol = R(y^2+1)
         Q4.<eye> = NumberField( pol ) # I already use i for other things
 
         # Factorize basis elements over Q4 and so on.
         d = []
         B = []
         h = []
         ideals_list = set()
         M = [] # Exponent matrix of the Bi's
 
         # Pre-process to find all ideals appearing in the factorization and fix
         # a chosen generator for each of them. This is important in order to
         # compute the "sign" (h-parameter) of an element with respect to it Bi.
         for g in b:
             factorization_list = list( Q4.ideal(g).factor() )
             ideals_list |= set( [ x[0] for x in factorization_list ] )
         ideals_list = list( ideals_list )
         # Chooses a generator of each principal ideal in the list
         irreducibles_list = [ J.gens_reduced()[0] for J in ideals_list ]
 
         # Compute the Q4-parameters of the given basis b. Also computes the
         # exponent matrix of the Bi's 
         for g in b:
             factorization_list = list( Q4.ideal(g).factor() )
             exps = [ x[1] for x in factorization_list ]
             d.append( divisibility( exps, l ) )
             Bg = 1
             for j in range(len(factorization_list)):
                 a = 0
                 for i in range( len( ideals_list ) ):
                     if ideals_list[i] == factorization_list[j][0]:
                         a = irreducibles_list[i]
                         break
                 Bg *= a ^ (exps[j]/(l^d[-1]))
             B.append(Bg)
             u = g / (Bg^(l^d[-1]))
             if not u.is_unit():
                 print( "Error: g is not the right power of the computed Bg." )
                 print( "g:", g, ", Bg:", Bg, ", exponent:", l^d[-1] )
             if u == 1:
                 h.append( 0 )
             elif u == -1:
                 h.append( 1 )
             else:
                 h.append( 2 ) 
 
         # Make the exponent matrix M (for now as a list of rows)
         for g in B:
             row = [0] * len(ideals_list)
             for i in range(len(ideals_list)):
                 I = ideals_list[i]
                 ee = 1
                 while (I^ee).divides(g):
                     ee += 1
                 row[i] = ee-1
             M.append(row)
 
         # If the Bi's are not strongly independent, apply the algorithm (only
         # once) to produce a new basis. The new basis has maximal parameters.
         coeffs = find_combination( matrix(M), l )
         if coeffs != []:
 
             maxi = -1
             maxd = -1
             for i in range(len(d)):
                 if d[i] > maxd and coeffs[i] != 0:
                     maxd = d[i]
                     maxi = i
 
             x = [(a/coeffs[maxi]).lift() for a in coeffs] # Now a vector of int
 
             new_element = 1
             for i in range(len(d)):
                 new_element *= b[i]^( x[i] * l^(d[maxi]-d[i]) )
         
             b[maxi] = new_element
 
             # Compute new B, d and so on.
             factorization_list = list( Q4.ideal(b[maxi]).factor() )
             exps = [ x[1] for x in factorization_list ]
             d[maxi] = divisibility( exps, l )
             Bg = 1
 
             for j in range(len(factorization_list)):
                 a = 0
                 for i in range( len( ideals_list ) ):
                     if ideals_list[i] == factorization_list[j][0]:
                         a = irreducibles_list[i]
                         break
                 Bg *= a ^ (exps[j]/(l^d[maxi]))
             B[maxi] = Bg
             M[maxi] = [ x[1] for x in list( Q4.ideal(Bg).factor() ) ]
             u = b[maxi] / (Bg^(l^d[maxi]))
             if not u.is_unit():
                 print( "Error: new element is not the right power of B." )
                 print( "New el.:",b[maxi],", B:",Bg,", exponent:",l^d[maxi] )
             if u == 1:
                 h[maxi] = 0
             elif u == -1:
                 h[maxi] = 1
             else:
                 h[maxi] = 2 
             
         return [(d[i],h[i]) for i in range(len(d))]
 
 # Uses Theorem 18 to compute the degree of Kummer extensions.
 def compute_vl( p, n, m, r ):
     h = [ x[1] for x in p ]
     ni = [ min( n, x[0] ) for x in p ]
     M = max( m, max( [ h[i] + ni[i] for i in range( len( p ) ) ] ) )
     
     return M - m + r*n - sum( ni )
 
 # Given any basis b of a group G computes an l-good basis for G. This is done
 # using the algorithm outlined in the proof of Theorem 14 of (Debry-Perucca).
 def make_good_basis( b, l ):
     M = exponent_matrix( b )
     d = []
     B = []
     for i in range(len(b)):
         di = divisibility( M[i], l )
         d.append( di )
         B.append( abs(b[i])^(1/(l^di)) )
 
     # Computes the coeffiecients of a linear combination of the rows of M
     # that is zero modulo l. These coefficients are elements of F_l.
     coeffs = find_combination( exponent_matrix( B ), l )
 
     while coeffs != []:
 
         # Computes which basis element (with non-zero coefficient in the linear
         # combination above) has maximal divisibility.
         maxi = -1
         maxd = -1
         for i in range(len(d)):
             if d[i] > maxd and coeffs[i] != 0:
                 maxd = d[i]
                 maxi = i
 
         x = [ (a/coeffs[maxi]).lift() for a in coeffs ] # Now a vector of ints
 
         new_element = 1
         for i in range(len(d)):
             new_element *= b[i]^( x[i] * l^(d[maxi]-d[i]) )
         
         b[maxi] = new_element
         M = exponent_matrix( b )
         d[maxi] = divisibility( M[maxi], l )
         B[maxi] = abs(b[maxi])^(1/(l^d[maxi]))
         
         coeffs = find_combination( exponent_matrix( B ), l )
 
     GB = [[]]
     for i in range(len(d)):
         while( len(GB) <= d[i] ):
             GB.append([])
         GB[d[i]].append(b[i])
     return GB
 
 # Takes a good basis B and adjusts the sign of the elements so that there is at
 # most one negative generator (of positive divisibility). The input is a good
 # basis in the format returned by make_good_basis.
 # Returns a pair (B,d), where B is the updated basis and d is the divisibility
 # parameter of the only negative element remained.
 # The sign of the d=0 elements is just ignored in the other steps of the
 # algorithm, so we keep them negative.
 def adjust_sign( B ):
     neg = 0
     dret = -1
     for d in range( len(B)-1, 0, -1 ):
         for i in range(len(B[d])):
             if B[d][i] < 0:
                 if neg == 0:
                     neg = B[d][i]
                     dret = d
                 else:
                     B[d][i] *= neg
     return (B,dret)
 
 # Given the exponent matrix M of a list of rational numbers B, returns the
 # coefficients of a linear combination that is weakly l-divisible, or [] if
 # the B[i] are strongly l-independent.
 def find_combination( M, l ):
     M = M.change_ring( GF( l ) )
     if M.rank() != min( M.dimensions() ):
         return M.kernel().basis()[0]
     else:
         return []
 
 # A rational number must be given as a list of exponents of primes in its
 # factorization.
 # Returns the minimal l-valuation of the exponents.
 def divisibility( A, l ):
     if len(A) == 0:
         print( "Warning: computing the divisibility of a torsion element.", end="" )
         print( "Returning +Infinity." )
         return +Infinity
     return min( [ valuation( x, l ) for x in A ] )
 
 # For a given set of non-zero rationals B, computes the "exponent matrix"
 def exponent_matrix( B ):
     prime_list = set()
     for g in B:
         prime_list |= set( prime_factors( g ) )
     prime_list = list( prime_list )
     np = len( prime_list )
     rows = []
     for g in B:
         rowg = [0] * np
         for f in list( factor( g ) ):
             for i in range( np ):
                 if f[0] == prime_list[i]:
                     rowg[i] = f[1]
         rows.append( rowg )
     return matrix( rows )
 
 # Computing the exponent matrix, but keeping an extra column for the signs.
 # Returns a pair (M,L) where M is the modified exponent matrix and L is the
 # list of primes appearing in the factorization.
 def exponent_matrix_with_sign_and_primes( B ): 
     prime_list = set()
     for g in B:
         prime_list |= set( prime_factors( g ) )
     prime_list = list( prime_list )
     np = len( prime_list )
     rows = []
     for g in B:
         rowg = [0] * np
         for f in list( factor( g ) ):
             for i in range( np ):
                 if f[0] == prime_list[i]:
                     rowg[i] = f[1]
         s = 0
         if sgn(g) == -1:
             s = 1
         rowg.append( s )
         rows.append( rowg )
     return ( matrix( rows ), prime_list )
 
 # This is a wrapper function for total_kummer_failure( G, True ), see below.
 def TotalKummerFailure( G ):
     total_kummer_failure( G, True )
 
 # Input: any set of generators for a subgroup G of Q*.
 # If output=False, returns a 4-uple (t,MM,NN,D):
 # - t is a pair, where t[0] is the rank of G and t[1] is either True (if G has
 #   torsion) or False (is it does not).
 # - MM is the pair (M0,divisors(M0))
 # - NN is the pair (N0,divisors(N0))
 # - D is a table F, where F[j][i] is the ration between phi(m)n^r and the
 #   degree of Q_{m,n} (i.e., the Kummer failure at m,n, where m is the i-th
 #   divisor of M0 and n is the j-th divisor of N0 (in the computed list))
 #   computed for a torsion-free part of G.
 # If output is True, outputs this data in a human-readable way and does not
 # return any value.
 def total_kummer_failure( G, output ):
 
     # Computing a basis.
     (BM,BM_primes) = exponent_matrix_with_sign_and_primes( G )
     BM = BM.echelon_form()
     BM_primes.append(-1)
     B = []
     torsion = False
 
     for r in BM.rows():
         gr = product( [ BM_primes[i]^r[i] for i in range(len(r)) ] )
         if gr == -1:
             torsion = True
             break
         elif gr == 1:
             break
         else:
             B.append(gr)
 
     if len(B) == 0:
         print( "G is torsion. The extension is cyclotomic. Stopping." )
         return False
 
     r = len(B) # Rank of G
 
     # Compute l-adic data (straightforward)
     ( bad_primes, l_adic_failure_table ) = total_l_adic_failure( B )
 
     # Compute adelic data.
     (GB,d) = adjust_sign( make_good_basis( B, 2 ) )
     adelic_failure_table = adelic_failure_gb( GB, d )
     
     # Computing the bounds M0 and N0
     N0 = 1
     for i in range(len( bad_primes )):
         N0 *= bad_primes[i] ^ len( l_adic_failure_table[i][-1][0] )
     # Extra factors of 2 may come from the adelic failure
     N0 = lcm( N0, 2^len( adelic_failure_table ) )
     divs_N0 = divisors(N0)
 
     # greatest M appearing in the adelic failure table
     M0 = adelic_failure_table[-1][-1][0]
     divs_M0 = divisors(M0)
 
     # Failure Table
     FT = [ [ 1 for d1 in divisors(M0) ] for d2 in divisors(N0) ]
 
     # Takes the adelic failure from the the relative table
     for i in range(1,len(adelic_failure_table)+1):
         for pp in adelic_failure_table[i-1]:
             # Runs through all divisors of M0 and N0 that may have this
             # failure (the table is not "complete").
             for l in range(len(divs_M0)):
                 for j in range(len(divs_N0)):
                     dM = divs_M0[l]
                     dN = divs_N0[j]
                     if dN%(2^i)==0:
                         if dN%(2^(i+1))!=0 or i==len(adelic_failure_table):
                             if dM % pp[0] == 0:  
                                 FT[j][l] = lcm( FT[j][l], pp[1] )
 
     # Adding l-adic failure to the table
     for i in range( len( bad_primes ) ):
         l = bad_primes[i]
         for j in range(len(divs_N0)):
             dN = divs_N0[j]
             fl = l_adic_failure_from_data(B,l,l_adic_failure_table[i],dN,dN)
             for h in range(len(divs_M0)):
                 FT[j][h] *= fl
 
     ret = ( ( r, torsion ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT )
 
     if output:
         print_total_table( ret )
         # Uncomment following line for case list description.
         #print_case_list( ret )
         return
     else:
         return ret
 
 # For the torsion tables, recall that when -1 is in G, the failure is defined
 # as the ration between 2^eN^r and the degree of Q_{M,N} over Q_M, where e=1
 # if N is even and e=0 otherwise.
 
 # Makes the failure table for the torsion case when M/N is even. In this case,
 # an entry of the table is doubled if the corresponding value of N (actually,
 # of gcd(N,N0) ) is even, and is kept the same otherwise.
 # The expected degree (over Q) 2^e * phi(M) * N^r, where e=1 if N is even and
 # e=0 otherwise.
 def torsion_table_even( data ):
 
     ( ( r, torsion ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT ) = data
 
     new_FT = [ [ 1 for d1 in divs_M0 ] for d2 in divs_N0 ]
     for i in range(len(divs_N0)):
         for j in range(len(divs_M0)):
             if divs_N0[i] % 2 == 0:
                 new_FT[i][j] = 2 * FT[i][j]
 
     return new_FT
 
 # Makes the failure table for the torsion case when M/N is odd. In this case
 # the entry at (M,N) is taken from the torsion-free entry at (2M,N).
 # In other words, the expected degree (over Q) 2^e * phi(M) * N^r, where e=1 if
 # N is even and e=0 otherwise.
 def torsion_table_odd( data ):
 
     ( ( r, torsion ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT ) = data
     
     new_FT = [ [ 1 for d1 in divs_M0 ] for d2 in divs_N0 ]
     new_data = ( ( r, False ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT )
     for i in range(len(divs_N0)):
         for j in range(len(divs_M0)):
             dN = divs_N0[i]
             dM = divs_M0[j]
             # The following lines just copy the failure from (2M,N) in the 
             # torsion-free case. It is easier to write it like this, although
             # it is not necessary in theory to compute the degree.
             exp_deg = euler_phi(2*dM) * dN^r
             deg = kummer_degree_from_total_table( 2*dM, dN, new_data ) 
             new_FT[i][j] = exp_deg / deg
     
     return new_FT
 
 def print_total_table( data ):
 
     ( ( r, torsion ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT ) = data
     
     print( "M_0 =", M0 )
     print( "N_0 =", N0 )
     print( "" )
     print( "The following table shows the total failure of Kummer degrees", end="" )
     if torsion:
         print( "in\n case the quotient M/N is EVEN." )
     else:
         print( "." )
     print( "Columns correspond to values of M, rows to values of N" )
     print( "" )
     print( "The degree of the Kummer extension (M,N) can be extracted by taking" )
     print( "the value f (failure) of the entry at (gcd(N,N0),gcd(M,M0)) and" )
     print( "simply computing ed(M,N) / f, where ed(M,N) is the expected degree" )
     print( "of the Kummer extension." )
     if torsion:
         print( "In this case (-1 is in G), we have ed(M,N) = 2^e*phi(M)*N^r," )
         print( "where e=1 if N is even and e=0 if N is odd." )
         FT1 = torsion_table_even( data )
     else:
         print( "In this case (G is torsion-free) we have ed(M,N) = phi(M)*N^r," )
         FT1 = FT
     print( "where r is the rank of G." )
     print( "" )
 
     tt = [ ["","|"] + divs_M0 ]
     tt.append( "-" * (len(divs_M0)+2) )
     for i in range(len(divs_N0)):
         tt.append( [ divs_N0[i] ] + ["|"]  + FT1[i] )
     print( table(tt) )
     print( "" )
 
     if torsion:
         print( "The following table shows the total failure of Kummer degrees in" )
         print( "case the quotient M/N is ODD." )
         print( "This table can be read exactly as the first one." )
         print( "" )
 
         # A good strategy is the following:
         #   A little translation exercise: the failure at (M,N) in the torsion
         #   case is either the same as that for (2M,N) in the torsion-free case
         #   (if M is even) or its double (if M is odd).
         # However, due to problems in reading the table for bigger M, it is
         # easier to just compute the degree every time, and then deduce the
         # failure. This is not too inefficient, since we can use the data that
         # we have already computed via kummer_degree_from_total_table.
         new_FT = torsion_table_odd( data )
         # Printing the new table
         tt = [ ["","|"] + divs_M0 ]
         tt.append( "-" * (len(divs_M0)+2) )
         for i in range(len(divs_N0)):
             tt.append( [ divs_N0[i] ] + ["|"]  + new_FT[i] )
         print( table(tt) )
         print( "" )
 
 def print_case_list( data ):
 
     ( ( r, torsion ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT ) = data
 
     FT_odd = torsion_table_odd( data )
     # FT1 is either FT or FT_even in the torsion case
     FT1 = FT
     if torsion:
         FT1 = torsion_table_even( data )
     pf = []
     for row in FT1:
         pf += row
     if torsion:
         for row in FT_odd:
             pf += row
     pf = list(set(pf))
     pf.sort()
     for f in pf:
         print( "Failure is", f, "if", end="" )
         if torsion:
             print( "M/N is EVEN and", end="" )
         print( "(gcd(M,M0),gcd(N,N0)) is one of the following:" )
         lijst = []
         for i in range(len(divs_N0)):
             for j in range(len(divs_M0)):
                 if FT1[i][j] == f:
                     lijst.append( ( divs_M0[j], divs_N0[i] ) )
         print( lijst )
         if torsion:
             print( "or if M/N is ODD and (gcd(M,M0)),gcd(N,N0)) is one of the", end="" )
             print( "following:" )
 
             lijst_odd = []
             for i in range(len(divs_N0)):
                 for j in range(len(divs_M0)):
                     if FT_odd[i][j] == f:
                         lijst_odd.append( ( divs_M0[j], divs_N0[i] ) )
             print( lijst_odd )
 
         print( "" )
 
 # Extracts a specific value of failure from the total table.
 def kummer_failure_from_total_table( M, N, data ):
     ( ( r, torsion ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT ) = data
     FT1 = FT
     if torsion:
         if (M/N) % 2 == 0:
             FT1 = torsion_table_even( data )
         else:
             FT1 = torsion_table_odd( data )
 
     i = divs_N0.index( gcd( N, N0 ) )
     j = divs_M0.index( gcd( M, M0 ) )
     return FT1[i][j]
 
 # Computes the degree of the Kummer extension (M,N), by taking as input the
 # table computed by TotalKummerFailure.
 def kummer_degree_from_total_table( M, N, data ):
     
     ( ( r, torsion ), ( M0, divs_M0 ), ( N0, divs_N0 ), FT ) = data
 
     exp_deg = euler_phi(M) * N^r
     if torsion and N % 2 == 0:
         exp_deg *= 2
     return exp_deg / kummer_failure_from_total_table( M, N, data )
 
 # Given a set of generators for a finitely generated subgroup G of the
 # multiplicative group of Q, returns the degree of Q_{M,N} over Q.
 # M must be a multiple of N.
 def KummerDegree( G, M, N ):
     if M % N != 0:
         print( "M is not a multiple of N" )
         return -1
 
     data = total_kummer_failure(G,False)
     ((r,torsion),(M0,divs_M0),(N0,divs_N0),FT) = data
 
     exp_deg = euler_phi(M) * N^r
     
     if torsion and N % 2 == 0:
         exp_deg *= 2
     
     j = divs_M0.index(gcd(M,M0))
     i = divs_N0.index(gcd(N,N0))
 
     if torsion: 
         if (M/N)%2 == 0:
             failure = torsion_table_even( data )
         else:
             failure = torsion_table_odd( data )
     else:
         failure = FT
 
     return exp_deg / failure[i][j]