############################################################################ # Calculation of Cross Functions and Four Factors for each Eckhart quintic # ############################################################################ # We first calculate the Cross functions as binomials in Yoshidas. We start with four expressions in the Yoshidas for the given Cross, and then use the action of W(E6) to find other equivalent expressions. We obtain 270 sets of 4 such expressions (135 up to sign) by grouping together binomials with the same expressions in ds. The result is a dictionary whose keys are the expressions in ds. They keys and the elements of each list lie in the Symbolic Ring. # We us this to create the dictionary of four factors for identifying the Cross functions (up to sign) with these sets. The keys of this dictionary are the anticanonical triangles (same keys as the Eckardt quintics). Each element of the dictionary is a list of products of 4 positive roots in E6. They keys and the elements of each list are elements on the ring S. # We then identify each one of these 45*3 = 135 expressions with a Cross function. The assignment is arbitrary (there is no canonical labeling for the Cross functions). # Using this labeling we identify the 4 binomials in the Yoshidas with a Crossi or -Crossi function. The result is a dictionary, where both the keys and the elements of each list lie in the Symbolic Ring. We use these functions and the saved output in the file "modified_linear_equations_XY.sage" # At the end of this file, we use the dictionary "cross_to_yoshidas.sobj" to find the linear and cubic equation we must input to "modify_linear_equations_XY.sage" and "modify_cubic_equation_XY.sage" ####################################################### # Setting up the ambient rings with the W(E6) action. # ####################################################### # A list of variables d_1, ..., d_6 for the coefficients ds = [var("d%s"%i) for i in range(1,7)] # A list of variables E_1, ..., E_6 Es = [var("E%s"%i) for i in range(1,7)] # A list of variables F_12, ..., F_56 Fs = [var("F%s%s"%(i,j)) for i in range(1,7) for j in range(1,7) if i < j] # A list of variables G_1, ..., G_6 Gs = [var("G%s"%i) for i in range(1,7)] # A list of variables X_12, ..., X_65 Xs = [var("X%s%s"%(i,j)) for i in range(1,7) for j in range(1,7) if i != j] # A list of variables Y_123456, ..., Y_162534 Ys = [var("Y%s%s%s%s%s%s"%(i,j,k,l,m,n)) for i in range(1,7) for j in range(1,7) for k in range(1,7) for l in range(1,7) for m in range(1,7) for n in range(1,7) if i