\\ Pari/GP code for working with number field 47.47.4663683520257583455651486936454104910113678905866717884728024389998128903778476103820811059427504675587221679201531780973276735641.1

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

\\ Define the number field: 
K = bnfinit(y^47 - y^46 - 322*y^45 + 257*y^44 + 46168*y^43 - 32856*y^42 - 3928972*y^41 + 2800261*y^40 + 223114373*y^39 - 173143135*y^38 - 9003548066*y^37 + 7929128719*y^36 + 268197678343*y^35 - 271080896060*y^34 - 6041385515832*y^33 + 6979624724342*y^32 + 104494842005772*y^31 - 136657122243604*y^30 - 1400247413062994*y^29 + 2052225750352966*y^28 + 14591458839594017*y^27 - 23782847219941819*y^26 - 118118630838181495*y^25 + 213321125182479282*y^24 + 738128822260196887*y^23 - 1480085655858130569*y^22 - 3514638820507932409*y^21 + 7908202660646656434*y^20 + 12455131093399060106*y^19 - 32248927636656562527*y^18 - 31433221457470396690*y^17 + 98930979031819081241*y^16 + 51111011671312688653*y^15 - 223422112969694652903*y^14 - 36045180999704955067*y^13 + 359687682365174498379*y^12 - 43326797472112521543*y^11 - 392741974421328427418*y^10 + 139313975200941570964*y^9 + 266729246622027266884*y^8 - 153819291119830123270*y^7 - 92300136797476126809*y^6 + 82337017997204992796*y^5 + 4438182302442319866*y^4 - 17703030829530739458*y^3 + 4375073965542285492*y^2 - 7552331260195558*y - 66410957360928749, 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^47 - x^46 - 322*x^45 + 257*x^44 + 46168*x^43 - 32856*x^42 - 3928972*x^41 + 2800261*x^40 + 223114373*x^39 - 173143135*x^38 - 9003548066*x^37 + 7929128719*x^36 + 268197678343*x^35 - 271080896060*x^34 - 6041385515832*x^33 + 6979624724342*x^32 + 104494842005772*x^31 - 136657122243604*x^30 - 1400247413062994*x^29 + 2052225750352966*x^28 + 14591458839594017*x^27 - 23782847219941819*x^26 - 118118630838181495*x^25 + 213321125182479282*x^24 + 738128822260196887*x^23 - 1480085655858130569*x^22 - 3514638820507932409*x^21 + 7908202660646656434*x^20 + 12455131093399060106*x^19 - 32248927636656562527*x^18 - 31433221457470396690*x^17 + 98930979031819081241*x^16 + 51111011671312688653*x^15 - 223422112969694652903*x^14 - 36045180999704955067*x^13 + 359687682365174498379*x^12 - 43326797472112521543*x^11 - 392741974421328427418*x^10 + 139313975200941570964*x^9 + 266729246622027266884*x^8 - 153819291119830123270*x^7 - 92300136797476126809*x^6 + 82337017997204992796*x^5 + 4438182302442319866*x^4 - 17703030829530739458*x^3 + 4375073965542285492*x^2 - 7552331260195558*x - 66410957360928749, 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]])