\\ Pari/GP code for working with number field 38.38.687661045808093482376579225097085116287670624782479569073265850359469722789446885178751177457664.1

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

\\ Define the number field: 
K = bnfinit(y^38 - 191*y^36 + 15662*y^34 - 730575*y^32 + 21705622*y^30 - 436357836*y^28 + 6157331558*y^26 - 62411853712*y^24 + 460944322269*y^22 - 2499589479956*y^20 + 9974553348782*y^18 - 29202012585177*y^16 + 62167403988174*y^14 - 94680464008195*y^12 + 100459885220950*y^10 - 71207905317537*y^8 + 31469275400290*y^6 - 7629511512714*y^4 + 744125807285*y^2 - 1101076991, 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 - 191*x^36 + 15662*x^34 - 730575*x^32 + 21705622*x^30 - 436357836*x^28 + 6157331558*x^26 - 62411853712*x^24 + 460944322269*x^22 - 2499589479956*x^20 + 9974553348782*x^18 - 29202012585177*x^16 + 62167403988174*x^14 - 94680464008195*x^12 + 100459885220950*x^10 - 71207905317537*x^8 + 31469275400290*x^6 - 7629511512714*x^4 + 744125807285*x^2 - 1101076991, 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]])