\\ Pari/GP code for working with number field 25.1.17183405982116876392097404040231169905747048161.1

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^25 + 18*y^23 - 17*y^22 + 163*y^21 + 778*y^20 + 668*y^19 + 6093*y^18 + 8161*y^17 - 16824*y^16 + 24958*y^15 + 156175*y^14 + 317021*y^13 + 1220510*y^12 + 3228424*y^11 + 5166827*y^10 + 7607147*y^9 + 10396892*y^8 + 8655374*y^7 + 3386053*y^6 + 971637*y^5 + 379214*y^4 - 252700*y^3 + 96379*y^2 - 14488*y + 1088, 1)

\\ Defining polynomial: 
K.pol

\\ Degree over Q: 
poldegree(K.pol)

\\ Signature: 
K.sign

\\ Discriminant: 
K.disc

\\ Ramified primes: 
factor(abs(K.disc))[,1]~

\\ Integral basis: 
K.zk

\\ Class group: 
K.clgp

\\ Unit rank: 
K.fu

\\ Generator for roots of unity: 
K.tu[2]

\\ Fundamental units: 
K.fu

\\ Regulator: 
K.reg

\\ Analytic class number formula: 
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^25 + 18*x^23 - 17*x^22 + 163*x^21 + 778*x^20 + 668*x^19 + 6093*x^18 + 8161*x^17 - 16824*x^16 + 24958*x^15 + 156175*x^14 + 317021*x^13 + 1220510*x^12 + 3228424*x^11 + 5166827*x^10 + 7607147*x^9 + 10396892*x^8 + 8655374*x^7 + 3386053*x^6 + 971637*x^5 + 379214*x^4 - 252700*x^3 + 96379*x^2 - 14488*x + 1088, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

\\ Intermediate fields: 
L = nfsubfields(K); L[2..length(b)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])