Magma V2.16-8 ######################################################### Checking Step A and Step C for the example of the paper ######################################################### %% One checks that the modified monodromy polynomial %% of the cover is irreducible, then one computes the %% valuation of the different ideal of L1/k. %% One obtains 30=(16-1)+(16-1). z2:=pAdicRing(2,1000); z2x:=PolynomialRing(z2); k:=TotallyRamifiedExtension(z2,x^5-2); ky:=PolynomialRing(k); c:=c0^4; d:=c0^6; f:=1+d*y^2+c*y^3+y^4+y^5; L:=(2*d*y+c*y^2+y^4)^4-2^6*f^3*(1+y)-c^2*2^4*f^2*y^2; F,s,C:=Factorization(L: Extensions :=true); L1:=C[1]`Extension; IsIrreducible(L); IsTotallyRamified(L1); Valuation(Discriminant(IntegerRing(L1),IntegerRing(k))); ######################################################### Checking Step F for the example of the paper ######################################################### %% Magma code written by J. Guardia, J. Montes and E. Nart. Z:=Integers(); ZX:=PolynomialRing(Integers()); QX:=PolynomialRing(Rationals()); K:=NumberField(X^5+2); c:=c0^4; d:=c0^6; Ky:=PolynomialRing(K); f:=1+d*y^2+c*y^3+y^4+y^5; s1:=2*d*y+c*y^2+y^4; L:=s1^4-2^6*f^3*(1+y)-c^2*y^2*16*f^2; M:=NumberField(L); MX:=PolynomialRing(M); FF:=T^2-(1+d*x^2+c*x^3+x^4+x^5); L1:=NumberField(FF); AL1:=AbsoluteField(L1); P:=DefiningPolynomial(AL1); PP:=P/Coefficient(P,160); PPP:=3^800*7^800*Evaluate(QX!PP,XX/3^5/7^5); P4:=ZX!PPP; S:=NumberField(P4); Attach("+Ideals.m"); Montes(2,S); [[P`e,P`f]:P in S`PrimeIdeals[2]]; s2:=S`LocalIndex[2]; dis2:=Valuation(Discriminant(P4),2); dis2-2*s2; ######################################################### Checking Step A and Step C for an example with p=3 ######################################################### z3:=pAdicRing(3,1000); z3x:=PolynomialRing(z3); k:=TotallyRamifiedExtension(z3,x^20-3); ky:=PolynomialRing(k); c:=pi^18; f:=1+c*y^4+y^9+y^10; s1:=c*y^3+y^9; L:=s1^9+3^12*f^8*(1+y)+(c*y)^3*(-3)^9*f^6; F,s,C:=Factorization(L: Extensions :=true); L1:=C[1]`Extension; IsIrreducible(L); IsTotallyRamified(L1); Valuation(Discriminant(IntegerRing(L1),IntegerRing(k)));