\\ Pari/GP code for working with number field 24.4.35863302598527284385359953566442393477815766818821430206298828125.2.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^24 - 8*y^23 + 64*y^22 - 902*y^21 + 6316*y^20 - 136934*y^19 + 609372*y^18 - 8671906*y^17 + 30971223*y^16 - 528120304*y^15 + 2227739352*y^14 - 27368555416*y^13 + 84384989853*y^12 - 980595596854*y^11 + 3158183643447*y^10 - 24849991956160*y^9 + 74624141823520*y^8 - 442422095928880*y^7 + 765727826720400*y^6 - 2316315944560320*y^5 - 3888629115012928*y^4 - 1094970506171776*y^3 - 29289495523457792*y^2 - 50029902292521984*y - 21580162302394368, 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^24 - 8*x^23 + 64*x^22 - 902*x^21 + 6316*x^20 - 136934*x^19 + 609372*x^18 - 8671906*x^17 + 30971223*x^16 - 528120304*x^15 + 2227739352*x^14 - 27368555416*x^13 + 84384989853*x^12 - 980595596854*x^11 + 3158183643447*x^10 - 24849991956160*x^9 + 74624141823520*x^8 - 442422095928880*x^7 + 765727826720400*x^6 - 2316315944560320*x^5 - 3888629115012928*x^4 - 1094970506171776*x^3 - 29289495523457792*x^2 - 50029902292521984*x - 21580162302394368, 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]])