\\ Pari/GP code for working with number field 38.0.47716070387122491878768794713669776106334434794159463718664766104756555702804321854228973388671875.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^38 - y^37 + 194*y^36 - 202*y^35 + 16279*y^34 - 7863*y^33 + 772861*y^32 + 402596*y^31 + 23045822*y^30 + 40384193*y^29 + 479536671*y^28 + 1351816180*y^27 + 7928555878*y^26 + 25902608554*y^25 + 111456458601*y^24 + 341385389946*y^23 + 1297163210108*y^22 + 3493633650369*y^21 + 12095301942774*y^20 + 29754895552994*y^19 + 90602135732080*y^18 + 209292172135889*y^17 + 578704163180687*y^16 + 1150796400064466*y^15 + 3249066298571457*y^14 + 4892053641982278*y^13 + 14078439508345248*y^12 + 18448183569455711*y^11 + 46083907583218775*y^10 + 65495333977938485*y^9 + 111449247373051429*y^8 + 177739871270855783*y^7 + 253288222342260757*y^6 + 334042708854918147*y^5 + 456173375797126253*y^4 + 247500381632980217*y^3 + 331215018154218380*y^2 + 303420818935344453*y + 79580728329881359, 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^38 - x^37 + 194*x^36 - 202*x^35 + 16279*x^34 - 7863*x^33 + 772861*x^32 + 402596*x^31 + 23045822*x^30 + 40384193*x^29 + 479536671*x^28 + 1351816180*x^27 + 7928555878*x^26 + 25902608554*x^25 + 111456458601*x^24 + 341385389946*x^23 + 1297163210108*x^22 + 3493633650369*x^21 + 12095301942774*x^20 + 29754895552994*x^19 + 90602135732080*x^18 + 209292172135889*x^17 + 578704163180687*x^16 + 1150796400064466*x^15 + 3249066298571457*x^14 + 4892053641982278*x^13 + 14078439508345248*x^12 + 18448183569455711*x^11 + 46083907583218775*x^10 + 65495333977938485*x^9 + 111449247373051429*x^8 + 177739871270855783*x^7 + 253288222342260757*x^6 + 334042708854918147*x^5 + 456173375797126253*x^4 + 247500381632980217*x^3 + 331215018154218380*x^2 + 303420818935344453*x + 79580728329881359, 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]])