\\ Pari/GP code for working with number field 42.0.2333538056680443170216809877092138324063458978840788486681774071870758537049156298101507.1

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

\\ Define the number field: 
K = bnfinit(y^42 - 11*y^41 + 31*y^40 + 15*y^39 + 149*y^38 - 2171*y^37 + 5622*y^36 - 5813*y^35 + 17703*y^34 - 77949*y^33 + 215171*y^32 - 500434*y^31 + 1078714*y^30 - 3010069*y^29 + 7913255*y^28 - 11776682*y^27 + 16679313*y^26 - 87643658*y^25 + 355007978*y^24 - 761687128*y^23 + 908129906*y^22 - 907038305*y^21 + 2012009714*y^20 - 6104907919*y^19 + 18723809190*y^18 - 48743446573*y^17 + 87856746678*y^16 - 105007899193*y^15 + 97597592982*y^14 - 99989052476*y^13 + 101465465536*y^12 - 67260548547*y^11 + 47339909902*y^10 - 82545626769*y^9 + 101683114718*y^8 - 83757841269*y^7 + 103258810875*y^6 - 130872100909*y^5 + 93586325459*y^4 - 47886183228*y^3 + 41077962943*y^2 - 30874848118*y + 9042834947, 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^42 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947, 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]])