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# Returns the l-dvisibility parameters of G over Q, given a good basis gb of G,
# as a list of pairs (di,hi).
def parameters_Q( gb, l ):
ret = []
for i in range( len( gb ) ):
for j in gb[i]:
ret.append( (i,0) )
return ret
# 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.
# If l=2, it uses the results of PST-2.
def parameters_Q4_new( gb, l ):
if l != 2:
return parameters_Q( gb, l )
else:
# Converts from "good basis format" to simple list
b = []
for x in gb:
b += x
M = exponent_matrix( b )
# Thanks to my terrible notation, the elements of b are what are called
# g_i in the article, while the elements of bb will be the b_i's.
bb = [ abs(b[i]) ^ (2^(-divisibility(M[i],2))) for i in range(len(b)) ]
# Computing a combination of bb elements of the form 2 * square.
MM = exponent_matrix( bb + [2] ).change_ring( GF( 2 ) )
for a in MM.kernel().basis():
if a[-1] != 0:
# the vector a[0:-1] gives the coefficients for a combination
# of the b_i's of the form 2 * square
# Basis elements that actually appear in the combination
c = [ b[i] for i in range(len(a[0:-1])) if a[i] != 0 ]
# Return the parameters, changing only the ones fo the element
# of highest divisibility that appears in the combination.
ret = []
div_max = -1
ind_max = -1
for i in range(len(b)):
ret.append( (divisibility(M[i],2), 1-max(0,sgn(b[i]))) )
if a[i] != 0 and ret[i][0] > div_max:
div_max = ret[i][0]
ind_max = i
d1, h1 = ret[ind_max]
if d1 == 1:
h1 = 1 - h1
elif d1 == 0:
h1 = 2
ret[ind_max] = ( d1+1, h1 )
return ret
return parameters_Q( gb, 2 )
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