\\ Pari/GP code for working with number field 29.1.75682241113219898520171301845005641468074963121.1

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

\\ Define the number field: 
K = bnfinit(y^29 - 9*y^28 + 56*y^27 - 308*y^26 + 1574*y^25 - 7243*y^24 + 28966*y^23 - 99448*y^22 + 293090*y^21 - 744900*y^20 + 1641343*y^19 - 3148829*y^18 + 5274672*y^17 - 7725970*y^16 + 9897023*y^15 - 11084677*y^14 + 10859104*y^13 - 9337717*y^12 + 7122047*y^11 - 4916951*y^10 + 3150459*y^9 - 1890622*y^8 + 1028653*y^7 - 470622*y^6 + 165361*y^5 - 41848*y^4 + 8324*y^3 - 1810*y^2 + 323*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^29 - 9*x^28 + 56*x^27 - 308*x^26 + 1574*x^25 - 7243*x^24 + 28966*x^23 - 99448*x^22 + 293090*x^21 - 744900*x^20 + 1641343*x^19 - 3148829*x^18 + 5274672*x^17 - 7725970*x^16 + 9897023*x^15 - 11084677*x^14 + 10859104*x^13 - 9337717*x^12 + 7122047*x^11 - 4916951*x^10 + 3150459*x^9 - 1890622*x^8 + 1028653*x^7 - 470622*x^6 + 165361*x^5 - 41848*x^4 + 8324*x^3 - 1810*x^2 + 323*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]])