\\ Pari/GP code for working with number field 45.45.40844008098536976898528926596491646392449447921859499771891413268750045487853347665634360111802937199321.1

\\ 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 - 126*y^43 + 350*y^42 + 7095*y^41 - 18207*y^40 - 237254*y^39 + 560949*y^38 + 5280957*y^37 - 11465163*y^36 - 83127609*y^35 + 164960568*y^34 + 959694733*y^33 - 1729668720*y^32 - 8316855543*y^31 + 13495839126*y^30 + 54911329452*y^29 - 79318322604*y^28 - 278624618803*y^27 + 353264598507*y^26 + 1090386630918*y^25 - 1193881051406*y^24 - 3286918329246*y^23 + 3054422504946*y^22 + 7584376524784*y^21 - 5885779838802*y^20 - 13239515051214*y^19 + 8489225140885*y^18 + 17170281215082*y^17 - 9118144480932*y^16 - 16115327765898*y^15 + 7281918623109*y^14 + 10535410010724*y^13 - 4323241543716*y^12 - 4518883017183*y^11 + 1877328461526*y^10 + 1138556965481*y^9 - 554295751305*y^8 - 124509763362*y^7 + 91825606123*y^6 - 3656082528*y^5 - 5240896776*y^4 + 961503906*y^3 - 1541688*y^2 - 11847522*y + 756289, 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 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289, 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]])