\\ Pari/GP code for working with number field 24.4.19756778413055716819205133752664064.2

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

\\ Define the number field: 
K = bnfinit(y^24 - 11*y^23 + 53*y^22 - 131*y^21 + 131*y^20 + 144*y^19 - 585*y^18 + 471*y^17 + 846*y^16 - 3226*y^15 + 6073*y^14 - 7616*y^13 + 4239*y^12 + 5146*y^11 - 14303*y^10 + 15630*y^9 - 9886*y^8 + 3829*y^7 - 1007*y^6 + 240*y^5 - 15*y^4 - 39*y^3 + 19*y^2 - 3*y + 1, 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^24 - 11*x^23 + 53*x^22 - 131*x^21 + 131*x^20 + 144*x^19 - 585*x^18 + 471*x^17 + 846*x^16 - 3226*x^15 + 6073*x^14 - 7616*x^13 + 4239*x^12 + 5146*x^11 - 14303*x^10 + 15630*x^9 - 9886*x^8 + 3829*x^7 - 1007*x^6 + 240*x^5 - 15*x^4 - 39*x^3 + 19*x^2 - 3*x + 1, 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]])