\\ Pari/GP code for working with number field 42.0.180282079628321418522579756639824623344453525380673158224385384254625263736344592716927.1

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

\\ Define the number field: 
K = bnfinit(y^42 - y^41 + 2*y^40 + 80*y^39 - 73*y^38 + 139*y^37 + 2635*y^36 - 2181*y^35 + 3934*y^34 + 46702*y^33 - 34779*y^32 + 59013*y^31 + 489577*y^30 - 325107*y^29 + 514966*y^28 + 3148122*y^27 - 1839378*y^26 + 2684924*y^25 + 12529404*y^24 - 6257436*y^23 + 8056643*y^22 + 30745048*y^21 - 12383236*y^20 + 11933159*y^19 + 46384476*y^18 - 13840937*y^17 + 3649186*y^16 + 41025120*y^15 - 8138959*y^14 - 9679906*y^13 + 28379769*y^12 - 8357477*y^11 + 3513515*y^10 + 45545418*y^9 - 12952087*y^8 + 8843411*y^7 + 29858262*y^6 - 8807791*y^5 + 14296232*y^4 - 4154414*y^3 - 181830*y^2 + 2389098*y + 733913, 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 - x^41 + 2*x^40 + 80*x^39 - 73*x^38 + 139*x^37 + 2635*x^36 - 2181*x^35 + 3934*x^34 + 46702*x^33 - 34779*x^32 + 59013*x^31 + 489577*x^30 - 325107*x^29 + 514966*x^28 + 3148122*x^27 - 1839378*x^26 + 2684924*x^25 + 12529404*x^24 - 6257436*x^23 + 8056643*x^22 + 30745048*x^21 - 12383236*x^20 + 11933159*x^19 + 46384476*x^18 - 13840937*x^17 + 3649186*x^16 + 41025120*x^15 - 8138959*x^14 - 9679906*x^13 + 28379769*x^12 - 8357477*x^11 + 3513515*x^10 + 45545418*x^9 - 12952087*x^8 + 8843411*x^7 + 29858262*x^6 - 8807791*x^5 + 14296232*x^4 - 4154414*x^3 - 181830*x^2 + 2389098*x + 733913, 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]])