\\ Pari/GP code for working with number field 42.0.16778739246697564329550246936340186720321059686137149293129858772813957238199098503.1

\\ 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 + 13*y^40 - 18*y^39 + 139*y^38 - 250*y^37 + 1451*y^36 + 321*y^35 + 11819*y^34 + 7069*y^33 + 101147*y^32 + 34723*y^31 + 906407*y^30 - 143662*y^29 + 2718926*y^28 - 434521*y^27 + 7093735*y^26 - 1731307*y^25 + 17818760*y^24 - 17036040*y^23 + 50814721*y^22 - 44435389*y^21 + 124550915*y^20 - 97201554*y^19 + 258557330*y^18 - 191477895*y^17 + 222793332*y^16 - 183429282*y^15 + 205234539*y^14 - 104619060*y^13 + 150123312*y^12 + 66144604*y^11 + 21062459*y^10 + 5929237*y^9 + 1586736*y^8 + 394344*y^7 + 106707*y^6 + 17635*y^5 + 2858*y^4 + 449*y^3 + 67*y^2 + 9*y + 1, 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 + 13*x^40 - 18*x^39 + 139*x^38 - 250*x^37 + 1451*x^36 + 321*x^35 + 11819*x^34 + 7069*x^33 + 101147*x^32 + 34723*x^31 + 906407*x^30 - 143662*x^29 + 2718926*x^28 - 434521*x^27 + 7093735*x^26 - 1731307*x^25 + 17818760*x^24 - 17036040*x^23 + 50814721*x^22 - 44435389*x^21 + 124550915*x^20 - 97201554*x^19 + 258557330*x^18 - 191477895*x^17 + 222793332*x^16 - 183429282*x^15 + 205234539*x^14 - 104619060*x^13 + 150123312*x^12 + 66144604*x^11 + 21062459*x^10 + 5929237*x^9 + 1586736*x^8 + 394344*x^7 + 106707*x^6 + 17635*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*x + 1, 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]])