\\ Pari/GP code for working with number field 26.2.2877467739962384875567767188720703125.1

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

\\ Define the number field: 
K = bnfinit(y^26 - 2*y^25 - 4*y^24 + 16*y^23 - 11*y^22 - 29*y^21 + 88*y^20 - 189*y^19 + 510*y^18 - 655*y^17 + 132*y^16 - 179*y^15 + 449*y^14 + 1090*y^13 + 114*y^12 - 781*y^11 - 543*y^10 - 58*y^9 + 245*y^8 + 173*y^7 + 196*y^6 + 159*y^5 + 34*y^4 - 4*y^3 + 6*y^2 - 6*y - 1, 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 - 2*x^25 - 4*x^24 + 16*x^23 - 11*x^22 - 29*x^21 + 88*x^20 - 189*x^19 + 510*x^18 - 655*x^17 + 132*x^16 - 179*x^15 + 449*x^14 + 1090*x^13 + 114*x^12 - 781*x^11 - 543*x^10 - 58*x^9 + 245*x^8 + 173*x^7 + 196*x^6 + 159*x^5 + 34*x^4 - 4*x^3 + 6*x^2 - 6*x - 1, 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]])