\\ Pari/GP code for working with number field 26.0.105838418476275527898387851073846090273368671.1

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

\\ Define the number field: 
K = bnfinit(y^26 - 4*y^25 + 29*y^24 - 114*y^23 + 437*y^22 - 1233*y^21 + 3738*y^20 - 6943*y^19 + 17106*y^18 - 22984*y^17 + 59870*y^16 - 97147*y^15 + 263098*y^14 - 450753*y^13 + 813670*y^12 - 1048444*y^11 + 1103470*y^10 - 985605*y^9 + 369592*y^8 - 235197*y^7 - 276734*y^6 + 186178*y^5 - 88862*y^4 + 240404*y^3 + 178609*y^2 + 96160*y + 61675, 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^26 - 4*x^25 + 29*x^24 - 114*x^23 + 437*x^22 - 1233*x^21 + 3738*x^20 - 6943*x^19 + 17106*x^18 - 22984*x^17 + 59870*x^16 - 97147*x^15 + 263098*x^14 - 450753*x^13 + 813670*x^12 - 1048444*x^11 + 1103470*x^10 - 985605*x^9 + 369592*x^8 - 235197*x^7 - 276734*x^6 + 186178*x^5 - 88862*x^4 + 240404*x^3 + 178609*x^2 + 96160*x + 61675, 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]])