\\ Pari/GP code for working with number field 39.39.8320218171789251028069979650899790239310515771557947367834077287023540263376505733290557577514801.2

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

\\ Define the number field: 
K = bnfinit(y^39 - 237*y^37 - 158*y^36 + 24885*y^35 + 31758*y^34 - 1521777*y^33 - 2797074*y^32 + 59981382*y^31 + 142319290*y^30 - 1593173961*y^29 - 4645709076*y^28 + 28932877445*y^27 + 102241716102*y^26 - 355443646614*y^25 - 1553454595933*y^24 + 2810177785458*y^23 + 16432247086167*y^22 - 11967377146298*y^21 - 120635759398347*y^20 - 1823113254630*y^19 + 605746157617644*y^18 + 341698986985275*y^17 - 2017784741258409*y^16 - 2057859861184228*y^15 + 4196664129873231*y^14 + 6285983642002179*y^13 - 4725650025499719*y^12 - 10914822027031086*y^11 + 1465701883750416*y^10 + 10558980638642454*y^9 + 2159465693586741*y^8 - 5293388896015467*y^7 - 2202937031211426*y^6 + 1157156267855625*y^5 + 698944716032751*y^4 - 59632171643934*y^3 - 73740757923315*y^2 - 5146532589771*y + 1075512295747, 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^39 - 237*x^37 - 158*x^36 + 24885*x^35 + 31758*x^34 - 1521777*x^33 - 2797074*x^32 + 59981382*x^31 + 142319290*x^30 - 1593173961*x^29 - 4645709076*x^28 + 28932877445*x^27 + 102241716102*x^26 - 355443646614*x^25 - 1553454595933*x^24 + 2810177785458*x^23 + 16432247086167*x^22 - 11967377146298*x^21 - 120635759398347*x^20 - 1823113254630*x^19 + 605746157617644*x^18 + 341698986985275*x^17 - 2017784741258409*x^16 - 2057859861184228*x^15 + 4196664129873231*x^14 + 6285983642002179*x^13 - 4725650025499719*x^12 - 10914822027031086*x^11 + 1465701883750416*x^10 + 10558980638642454*x^9 + 2159465693586741*x^8 - 5293388896015467*x^7 - 2202937031211426*x^6 + 1157156267855625*x^5 + 698944716032751*x^4 - 59632171643934*x^3 - 73740757923315*x^2 - 5146532589771*x + 1075512295747, 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]])