\\ Pari/GP code for working with number field 29.1.77103436114042117740511038546742321228145336964361.1

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

\\ Define the number field: 
K = bnfinit(y^29 - 7*y^28 + 47*y^27 - 186*y^26 + 710*y^25 - 970*y^24 + 1144*y^23 + 2298*y^22 - 8465*y^21 + 16505*y^20 - 20589*y^19 + 3590*y^18 + 30661*y^17 - 81803*y^16 + 117497*y^15 - 94780*y^14 + 10260*y^13 + 127678*y^12 - 249606*y^11 + 245472*y^10 - 99463*y^9 - 191187*y^8 + 582067*y^7 - 785770*y^6 + 915704*y^5 - 793384*y^4 + 541792*y^3 - 324448*y^2 + 144320*y - 23552, 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 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552, 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]])