\\ Pari/GP code for working with number field 38.0.15223168714879313546692095257379448420591016496978036941227483116202289492875331751113046747.1.
\\ 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 + 91*y^36 - 24*y^35 + 5113*y^34 - 166*y^33 + 173717*y^32 + 27194*y^31 + 4239797*y^30 + 1325317*y^29 + 74027272*y^28 + 40999344*y^27 + 983326589*y^26 + 743120555*y^25 + 10059741791*y^24 + 9563481092*y^23 + 81070213075*y^22 + 87547591746*y^21 + 511882174610*y^20 + 601337789804*y^19 + 2539884758984*y^18 + 3042827513863*y^17 + 9675050790560*y^16 + 11543123809355*y^15 + 28268796522511*y^14 + 31776668507252*y^13 + 60600608962241*y^12 + 63076854532617*y^11 + 95016529831875*y^10 + 85766481354393*y^9 + 98440678914956*y^8 + 74640315704381*y^7 + 67352150639088*y^6 + 40110464366364*y^5 + 23287561297245*y^4 + 7350839404496*y^3 + 1810153278801*y^2 + 182761486103*y + 13841287201, 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 + 91*x^36 - 24*x^35 + 5113*x^34 - 166*x^33 + 173717*x^32 + 27194*x^31 + 4239797*x^30 + 1325317*x^29 + 74027272*x^28 + 40999344*x^27 + 983326589*x^26 + 743120555*x^25 + 10059741791*x^24 + 9563481092*x^23 + 81070213075*x^22 + 87547591746*x^21 + 511882174610*x^20 + 601337789804*x^19 + 2539884758984*x^18 + 3042827513863*x^17 + 9675050790560*x^16 + 11543123809355*x^15 + 28268796522511*x^14 + 31776668507252*x^13 + 60600608962241*x^12 + 63076854532617*x^11 + 95016529831875*x^10 + 85766481354393*x^9 + 98440678914956*x^8 + 74640315704381*x^7 + 67352150639088*x^6 + 40110464366364*x^5 + 23287561297245*x^4 + 7350839404496*x^3 + 1810153278801*x^2 + 182761486103*x + 13841287201, 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]])