\\ Pari/GP code for working with number field 38.38.249822358047761737585176935673663749247824265938007663448506628820714951323582836933135986328125.1

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

\\ Define the number field: 
K = bnfinit(y^38 - 17*y^37 - 64*y^36 + 2589*y^35 - 5411*y^34 - 156768*y^33 + 733103*y^32 + 4788192*y^31 - 35074270*y^30 - 69778288*y^29 + 937716801*y^28 - 3988006*y^27 - 15662818901*y^26 + 18940301798*y^25 + 169186329709*y^24 - 366424944806*y^23 - 1167394423359*y^22 + 3780063487716*y^21 + 4693995261962*y^20 - 24515638087910*y^19 - 6315614627194*y^18 + 103828189076722*y^17 - 36155927083660*y^16 - 285274970106320*y^15 + 223944436674986*y^14 + 486329219734937*y^13 - 571966582999537*y^12 - 458764738319995*y^11 + 781485719820911*y^10 + 160103399248715*y^9 - 557488639455834*y^8 + 53135732479684*y^7 + 180915511892981*y^6 - 38184876354047*y^5 - 25486623807243*y^4 + 6173666606950*y^3 + 1214949029058*y^2 - 273982926995*y + 184740541, 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^38 - 17*x^37 - 64*x^36 + 2589*x^35 - 5411*x^34 - 156768*x^33 + 733103*x^32 + 4788192*x^31 - 35074270*x^30 - 69778288*x^29 + 937716801*x^28 - 3988006*x^27 - 15662818901*x^26 + 18940301798*x^25 + 169186329709*x^24 - 366424944806*x^23 - 1167394423359*x^22 + 3780063487716*x^21 + 4693995261962*x^20 - 24515638087910*x^19 - 6315614627194*x^18 + 103828189076722*x^17 - 36155927083660*x^16 - 285274970106320*x^15 + 223944436674986*x^14 + 486329219734937*x^13 - 571966582999537*x^12 - 458764738319995*x^11 + 781485719820911*x^10 + 160103399248715*x^9 - 557488639455834*x^8 + 53135732479684*x^7 + 180915511892981*x^6 - 38184876354047*x^5 - 25486623807243*x^4 + 6173666606950*x^3 + 1214949029058*x^2 - 273982926995*x + 184740541, 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]])