\\ Pari/GP code for working with number field 44.0.361358208821422691858000111178424879775250540561407860223115640123091356736736779241.1

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

\\ Define the number field: 
K = bnfinit(y^44 - y^43 - 31*y^42 + 18*y^41 + 509*y^40 - 128*y^39 - 6323*y^38 + 711*y^37 + 64913*y^36 - 2855*y^35 - 548822*y^34 - 30450*y^33 + 3909507*y^32 + 497028*y^31 - 24053724*y^30 - 3501489*y^29 + 127000724*y^28 + 22049413*y^27 - 570173510*y^26 - 120800376*y^25 + 2191193723*y^24 + 441547756*y^23 - 7154579015*y^22 - 1235011630*y^21 + 19357719068*y^20 + 3692553824*y^19 - 42800540592*y^18 - 8560769600*y^17 + 77181753152*y^16 + 8780862080*y^15 - 109191557376*y^14 - 4336701952*y^13 + 115169783808*y^12 + 13857435648*y^11 - 86371995648*y^10 - 18961448960*y^9 + 47011856384*y^8 + 1711374336*y^7 - 14905901056*y^6 + 3448373248*y^5 + 3020947456*y^4 - 373293056*y^3 - 84934656*y^2 - 12582912*y + 4194304, 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 - x^43 - 31*x^42 + 18*x^41 + 509*x^40 - 128*x^39 - 6323*x^38 + 711*x^37 + 64913*x^36 - 2855*x^35 - 548822*x^34 - 30450*x^33 + 3909507*x^32 + 497028*x^31 - 24053724*x^30 - 3501489*x^29 + 127000724*x^28 + 22049413*x^27 - 570173510*x^26 - 120800376*x^25 + 2191193723*x^24 + 441547756*x^23 - 7154579015*x^22 - 1235011630*x^21 + 19357719068*x^20 + 3692553824*x^19 - 42800540592*x^18 - 8560769600*x^17 + 77181753152*x^16 + 8780862080*x^15 - 109191557376*x^14 - 4336701952*x^13 + 115169783808*x^12 + 13857435648*x^11 - 86371995648*x^10 - 18961448960*x^9 + 47011856384*x^8 + 1711374336*x^7 - 14905901056*x^6 + 3448373248*x^5 + 3020947456*x^4 - 373293056*x^3 - 84934656*x^2 - 12582912*x + 4194304, 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]])