\\ Pari/GP code for working with number field 44.0.637355869329332251201071317246647563330929205499488362752542693433916506945636309928088212472341047144500892270592.1

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

\\ Define the number field: 
K = bnfinit(y^44 - 4*y^43 + 450*y^42 - 1696*y^41 + 95623*y^40 - 339364*y^39 + 12725514*y^38 - 42479344*y^37 + 1186619666*y^36 - 3719731600*y^35 + 82234067400*y^34 - 241573595600*y^33 + 4387101211849*y^32 - 12047214773252*y^31 + 184321188359002*y^30 - 471779938027168*y^29 + 6195716143526136*y^28 - 14733546436300584*y^27 + 168602100536112036*y^26 - 371193178297827952*y^25 + 3750658087111474564*y^24 - 7615609064474320416*y^23 + 68785024359507662624*y^22 - 128259317837580249532*y^21 + 1047347442861854229432*y^20 - 1784203887396236578908*y^19 + 13304629870600067621984*y^18 - 20569608214806383894840*y^17 + 141251615396772952843075*y^16 - 196435665083615705294760*y^15 + 1250953936009105860804974*y^14 - 1545994678195125737706904*y^13 + 9182221521270632077265647*y^12 - 9915451857640942707235084*y^11 + 55177014296826057984352506*y^10 - 50811367810969170459257440*y^9 + 265871502086628031896736617*y^8 - 201251578078840046061089680*y^7 + 992905136657073086108322990*y^6 - 581679407650510662543408668*y^5 + 2711240748706355837313730686*y^4 - 1097811866899303633376184156*y^3 + 4840346129829289788210382520*y^2 - 1021208505276065935211161352*y + 4261581640413011116141369057, 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^44 - 4*x^43 + 450*x^42 - 1696*x^41 + 95623*x^40 - 339364*x^39 + 12725514*x^38 - 42479344*x^37 + 1186619666*x^36 - 3719731600*x^35 + 82234067400*x^34 - 241573595600*x^33 + 4387101211849*x^32 - 12047214773252*x^31 + 184321188359002*x^30 - 471779938027168*x^29 + 6195716143526136*x^28 - 14733546436300584*x^27 + 168602100536112036*x^26 - 371193178297827952*x^25 + 3750658087111474564*x^24 - 7615609064474320416*x^23 + 68785024359507662624*x^22 - 128259317837580249532*x^21 + 1047347442861854229432*x^20 - 1784203887396236578908*x^19 + 13304629870600067621984*x^18 - 20569608214806383894840*x^17 + 141251615396772952843075*x^16 - 196435665083615705294760*x^15 + 1250953936009105860804974*x^14 - 1545994678195125737706904*x^13 + 9182221521270632077265647*x^12 - 9915451857640942707235084*x^11 + 55177014296826057984352506*x^10 - 50811367810969170459257440*x^9 + 265871502086628031896736617*x^8 - 201251578078840046061089680*x^7 + 992905136657073086108322990*x^6 - 581679407650510662543408668*x^5 + 2711240748706355837313730686*x^4 - 1097811866899303633376184156*x^3 + 4840346129829289788210382520*x^2 - 1021208505276065935211161352*x + 4261581640413011116141369057, 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]])