\\ Pari/GP code for working with number field 27.27.124252631053426325344275434705435089635453266149328961.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^27 - 117*y^25 - 72*y^24 + 5508*y^23 + 7281*y^22 - 134253*y^21 - 286200*y^20 + 1772163*y^19 + 5648910*y^18 - 10963863*y^17 - 59816781*y^16 - 2220768*y^15 + 324023922*y^14 + 421698816*y^13 - 649983591*y^12 - 2060278281*y^11 - 956349270*y^10 + 2816524353*y^9 + 4662543087*y^8 + 1708361622*y^7 - 2605453038*y^6 - 3926837340*y^5 - 2503473813*y^4 - 891775287*y^3 - 177744726*y^2 - 17712225*y - 675379, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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^27 - 117*x^25 - 72*x^24 + 5508*x^23 + 7281*x^22 - 134253*x^21 - 286200*x^20 + 1772163*x^19 + 5648910*x^18 - 10963863*x^17 - 59816781*x^16 - 2220768*x^15 + 324023922*x^14 + 421698816*x^13 - 649983591*x^12 - 2060278281*x^11 - 956349270*x^10 + 2816524353*x^9 + 4662543087*x^8 + 1708361622*x^7 - 2605453038*x^6 - 3926837340*x^5 - 2503473813*x^4 - 891775287*x^3 - 177744726*x^2 - 17712225*x - 675379, 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(L)]

\\ Galois group: 
polgalois(K.pol)

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