\\ Pari/GP code for working with number field 45.15.22655076666278607886805489601267939337377890930813594593620979789073992426604317061891.1

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

\\ Define the number field: 
K = bnfinit(y^45 - 3*y^44 + 19*y^43 - 46*y^42 + 57*y^41 - 164*y^40 - 1542*y^39 + 2752*y^38 - 13177*y^37 + 27636*y^36 - 5513*y^35 + 194674*y^34 - 95004*y^33 + 1568504*y^32 - 4411428*y^31 + 7701411*y^30 - 22788169*y^29 + 356245*y^28 + 13001022*y^27 - 120681074*y^26 + 379138834*y^25 - 251876102*y^24 + 576442931*y^23 + 394617661*y^22 - 1914308187*y^21 + 1873927920*y^20 - 5326366508*y^19 + 1371219529*y^18 + 2001539759*y^17 - 9889048234*y^16 + 22330230077*y^15 - 9918095549*y^14 + 1934927850*y^13 + 41001461054*y^12 - 37979697765*y^11 + 8169529015*y^10 + 13175783912*y^9 - 55418628407*y^8 + 2188320806*y^7 - 1793697518*y^6 - 21729183123*y^5 - 927541692*y^4 + 301468299*y^3 + 1484196634*y^2 + 59977427*y - 879691, 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^45 - 3*x^44 + 19*x^43 - 46*x^42 + 57*x^41 - 164*x^40 - 1542*x^39 + 2752*x^38 - 13177*x^37 + 27636*x^36 - 5513*x^35 + 194674*x^34 - 95004*x^33 + 1568504*x^32 - 4411428*x^31 + 7701411*x^30 - 22788169*x^29 + 356245*x^28 + 13001022*x^27 - 120681074*x^26 + 379138834*x^25 - 251876102*x^24 + 576442931*x^23 + 394617661*x^22 - 1914308187*x^21 + 1873927920*x^20 - 5326366508*x^19 + 1371219529*x^18 + 2001539759*x^17 - 9889048234*x^16 + 22330230077*x^15 - 9918095549*x^14 + 1934927850*x^13 + 41001461054*x^12 - 37979697765*x^11 + 8169529015*x^10 + 13175783912*x^9 - 55418628407*x^8 + 2188320806*x^7 - 1793697518*x^6 - 21729183123*x^5 - 927541692*x^4 + 301468299*x^3 + 1484196634*x^2 + 59977427*x - 879691, 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]])