\\ Pari/GP code for working with number field 42.0.2281836760183646137444154412268560109828024514076489472840222217265158917203.1

\\ 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 + 21*y^40 - 18*y^39 + 248*y^38 - 191*y^37 + 1996*y^36 - 1375*y^35 + 12088*y^34 - 7495*y^33 + 57373*y^32 - 31825*y^31 + 219409*y^30 - 108700*y^29 + 684645*y^28 - 300153*y^27 + 1757705*y^26 - 677756*y^25 + 3715466*y^24 - 1242500*y^23 + 6455978*y^22 - 1853597*y^21 + 9148985*y^20 - 2205194*y^19 + 10469894*y^18 - 2090336*y^17 + 9505112*y^16 - 1507895*y^15 + 6709547*y^14 - 839645*y^13 + 3559478*y^12 - 314951*y^11 + 1368367*y^10 - 93808*y^9 + 355278*y^8 - 10362*y^7 + 58036*y^6 - 2497*y^5 + 4950*y^4 + 165*y^3 + 176*y^2 - 11*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 + 21*x^40 - 18*x^39 + 248*x^38 - 191*x^37 + 1996*x^36 - 1375*x^35 + 12088*x^34 - 7495*x^33 + 57373*x^32 - 31825*x^31 + 219409*x^30 - 108700*x^29 + 684645*x^28 - 300153*x^27 + 1757705*x^26 - 677756*x^25 + 3715466*x^24 - 1242500*x^23 + 6455978*x^22 - 1853597*x^21 + 9148985*x^20 - 2205194*x^19 + 10469894*x^18 - 2090336*x^17 + 9505112*x^16 - 1507895*x^15 + 6709547*x^14 - 839645*x^13 + 3559478*x^12 - 314951*x^11 + 1368367*x^10 - 93808*x^9 + 355278*x^8 - 10362*x^7 + 58036*x^6 - 2497*x^5 + 4950*x^4 + 165*x^3 + 176*x^2 - 11*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]])