\\ Pari/GP code for working with number field 45.45.18385118110130638532679220392530019242562666485797771487713623864599015675873914118339381361.1

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

\\ Define the number field: 
K = bnfinit(y^45 - 87*y^43 - 4*y^42 + 3402*y^41 + 300*y^40 - 79106*y^39 - 10008*y^38 + 1219668*y^37 + 196264*y^36 - 13165542*y^35 - 2519472*y^34 + 102436164*y^33 + 22316760*y^32 - 583045167*y^31 - 140145264*y^30 + 2440735434*y^29 + 631757436*y^28 - 7504722910*y^27 - 2050106328*y^26 + 16845457116*y^25 + 4771066968*y^24 - 27354225744*y^23 - 7899876240*y^22 + 31819110888*y^21 + 9218870784*y^20 - 26289431304*y^19 - 7517163552*y^18 + 15324352272*y^17 + 4245866400*y^16 - 6258116631*y^15 - 1643669037*y^14 + 1770836592*y^13 + 429144192*y^12 - 340508538*y^11 - 73586928*y^10 + 43022076*y^9 + 7934841*y^8 - 3375522*y^7 - 502012*y^6 + 149688*y^5 + 16758*y^4 - 3200*y^3 - 252*y^2 + 24*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^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*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]])