\\ Pari/GP code for working with number field 45.45.202328376069000568231882298692586415768387771371083469740034658547467305567124640219844877719879150390625.1

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

\\ Define the number field: 
K = bnfinit(y^45 - 165*y^43 - 25*y^42 + 12060*y^41 + 3486*y^40 - 519330*y^39 - 214140*y^38 + 14776560*y^37 + 7718745*y^36 - 295077669*y^35 - 182852610*y^34 + 4284179615*y^33 + 3016157820*y^32 - 46197393465*y^31 - 35772719505*y^30 + 374607222195*y^29 + 310484223975*y^28 - 2298730688540*y^27 - 1988244775980*y^26 + 10695933805596*y^25 + 9407317026635*y^24 - 37696094037390*y^23 - 32764647222885*y^22 + 100289481110340*y^21 + 83332538318277*y^20 - 200279250507195*y^19 - 152889355168310*y^18 + 297466429398705*y^17 + 198745289480940*y^16 - 323624520119411*y^15 - 178210655500665*y^14 + 251547979741875*y^13 + 105785393431525*y^12 - 134218212107550*y^11 - 38927292179565*y^10 + 46104688292390*y^9 + 7956837983055*y^8 - 9173261413650*y^7 - 765305596255*y^6 + 891053703312*y^5 + 40709257785*y^4 - 38148076985*y^3 - 1973427960*y^2 + 606052050*y + 47691757, 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 - 165*x^43 - 25*x^42 + 12060*x^41 + 3486*x^40 - 519330*x^39 - 214140*x^38 + 14776560*x^37 + 7718745*x^36 - 295077669*x^35 - 182852610*x^34 + 4284179615*x^33 + 3016157820*x^32 - 46197393465*x^31 - 35772719505*x^30 + 374607222195*x^29 + 310484223975*x^28 - 2298730688540*x^27 - 1988244775980*x^26 + 10695933805596*x^25 + 9407317026635*x^24 - 37696094037390*x^23 - 32764647222885*x^22 + 100289481110340*x^21 + 83332538318277*x^20 - 200279250507195*x^19 - 152889355168310*x^18 + 297466429398705*x^17 + 198745289480940*x^16 - 323624520119411*x^15 - 178210655500665*x^14 + 251547979741875*x^13 + 105785393431525*x^12 - 134218212107550*x^11 - 38927292179565*x^10 + 46104688292390*x^9 + 7956837983055*x^8 - 9173261413650*x^7 - 765305596255*x^6 + 891053703312*x^5 + 40709257785*x^4 - 38148076985*x^3 - 1973427960*x^2 + 606052050*x + 47691757, 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]])