\\ Pari/GP code for working with number field 29.29.1931816133436230496253440348173909042983780087233421135736506627041.1

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

\\ Define the number field: 
K = bnfinit(y^29 - y^28 - 112*y^27 + 91*y^26 + 5198*y^25 - 3644*y^24 - 132219*y^23 + 83238*y^22 + 2053518*y^21 - 1187959*y^20 - 20553532*y^19 + 11071128*y^18 + 136460842*y^17 - 69042962*y^16 - 609473492*y^15 + 292259011*y^14 + 1836592125*y^13 - 845018358*y^12 - 3706016039*y^11 + 1661552324*y^10 + 4906886664*y^9 - 2177019390*y^8 - 4095369839*y^7 + 1819962089*y^6 + 1998032360*y^5 - 895362174*y^4 - 490947342*y^3 + 221892059*y^2 + 42927079*y - 19524467, 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^29 - x^28 - 112*x^27 + 91*x^26 + 5198*x^25 - 3644*x^24 - 132219*x^23 + 83238*x^22 + 2053518*x^21 - 1187959*x^20 - 20553532*x^19 + 11071128*x^18 + 136460842*x^17 - 69042962*x^16 - 609473492*x^15 + 292259011*x^14 + 1836592125*x^13 - 845018358*x^12 - 3706016039*x^11 + 1661552324*x^10 + 4906886664*x^9 - 2177019390*x^8 - 4095369839*x^7 + 1819962089*x^6 + 1998032360*x^5 - 895362174*x^4 - 490947342*x^3 + 221892059*x^2 + 42927079*x - 19524467, 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]])