\\ Pari/GP code for working with number field 24.4.896582564963182109633998839161059836945394170470535755157470703125.6.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^24 - 4*y^23 - 99*y^22 + 431*y^21 - 14784*y^20 - 3090*y^19 + 940100*y^18 - 2629940*y^17 + 74365635*y^16 + 214136855*y^15 - 2267018532*y^14 + 877025663*y^13 - 159570300332*y^12 - 33238859092*y^11 + 1688439158893*y^10 + 2467042788125*y^9 + 87026862121385*y^8 - 19654322538325*y^7 + 696572665120450*y^6 + 4771707259687380*y^5 - 7522754457264744*y^4 + 27858329958403216*y^3 + 49846168997904256*y^2 - 92937065600889664*y + 150093087495580416, 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 - 4*x^23 - 99*x^22 + 431*x^21 - 14784*x^20 - 3090*x^19 + 940100*x^18 - 2629940*x^17 + 74365635*x^16 + 214136855*x^15 - 2267018532*x^14 + 877025663*x^13 - 159570300332*x^12 - 33238859092*x^11 + 1688439158893*x^10 + 2467042788125*x^9 + 87026862121385*x^8 - 19654322538325*x^7 + 696572665120450*x^6 + 4771707259687380*x^5 - 7522754457264744*x^4 + 27858329958403216*x^3 + 49846168997904256*x^2 - 92937065600889664*x + 150093087495580416, 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]])