\\ Pari/GP code for working with number field 26.0.549596338332815511233443533045654296875.1

\\ 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 + 5*y^24 - 26*y^23 + 53*y^22 - 54*y^21 + 268*y^20 - 237*y^19 + 744*y^18 + 440*y^17 + 1338*y^16 - 3330*y^15 - 8756*y^14 + 7284*y^13 + 63381*y^12 + 167449*y^11 + 262693*y^10 + 320455*y^9 + 419505*y^8 + 480606*y^7 + 569660*y^6 + 459537*y^5 + 397335*y^4 + 258066*y^3 + 85342*y^2 + 34263*y + 39371, 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 + 5*x^24 - 26*x^23 + 53*x^22 - 54*x^21 + 268*x^20 - 237*x^19 + 744*x^18 + 440*x^17 + 1338*x^16 - 3330*x^15 - 8756*x^14 + 7284*x^13 + 63381*x^12 + 167449*x^11 + 262693*x^10 + 320455*x^9 + 419505*x^8 + 480606*x^7 + 569660*x^6 + 459537*x^5 + 397335*x^4 + 258066*x^3 + 85342*x^2 + 34263*x + 39371, 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]])