\\ Pari/GP code for working with number field 38.0.10517653274833793910788391650485470831673313848992580618621786800117331392788597759645606697493139.1.
\\ 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 + 6*y^36 - 46*y^35 - 866*y^34 - 2262*y^33 - 8218*y^32 - 21902*y^31 + 431634*y^30 + 845654*y^29 + 12392062*y^28 + 22625646*y^27 + 131347110*y^26 + 99749222*y^25 + 35665294*y^24 - 866315187*y^23 - 258426421*y^22 + 17785842992*y^21 + 90256146871*y^20 + 153552360949*y^19 + 435041961961*y^18 - 1392669637447*y^17 + 1076840441256*y^16 - 9935143209215*y^15 + 31070017846749*y^14 - 17675958023404*y^13 + 138250934902501*y^12 - 327233950365006*y^11 + 433153745352823*y^10 - 1944991946784557*y^9 + 5889532748622531*y^8 - 11577094600459131*y^7 + 25495193933767723*y^6 - 50712315827213047*y^5 + 74651226530405679*y^4 - 82308207923451862*y^3 + 77528185035084423*y^2 - 59368882440546583*y + 25758699005655811, 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 + 6*x^36 - 46*x^35 - 866*x^34 - 2262*x^33 - 8218*x^32 - 21902*x^31 + 431634*x^30 + 845654*x^29 + 12392062*x^28 + 22625646*x^27 + 131347110*x^26 + 99749222*x^25 + 35665294*x^24 - 866315187*x^23 - 258426421*x^22 + 17785842992*x^21 + 90256146871*x^20 + 153552360949*x^19 + 435041961961*x^18 - 1392669637447*x^17 + 1076840441256*x^16 - 9935143209215*x^15 + 31070017846749*x^14 - 17675958023404*x^13 + 138250934902501*x^12 - 327233950365006*x^11 + 433153745352823*x^10 - 1944991946784557*x^9 + 5889532748622531*x^8 - 11577094600459131*x^7 + 25495193933767723*x^6 - 50712315827213047*x^5 + 74651226530405679*x^4 - 82308207923451862*x^3 + 77528185035084423*x^2 - 59368882440546583*x + 25758699005655811, 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]])