From 6b80453f9f7cd302899ee62eb068d1bfe0bc9ba5 Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Tue, 17 Sep 2019 10:29:15 +0200 Subject: Removed some commented code, changed README.md --- parameters_Q4_old.sage | 123 ------------------------------------------------- 1 file changed, 123 deletions(-) delete mode 100644 parameters_Q4_old.sage (limited to 'parameters_Q4_old.sage') diff --git a/parameters_Q4_old.sage b/parameters_Q4_old.sage deleted file mode 100644 index d96fd8c..0000000 --- a/parameters_Q4_old.sage +++ /dev/null @@ -1,123 +0,0 @@ -# Computes the l-divisibility parameters of G over Q4, given a good basis gb -# 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. = PolynomialRing( QQ ) - pol = R(y^2+1) - Q4. = 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))] - - -- cgit v1.3