\\ Pari/GP code for working with number field 42.42.565343212441678035532894502003808167878401992443661947648452445739810658542578516149.1

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

\\ Define the number field: 
K = bnfinit(y^42 - 81*y^40 - 4*y^39 + 2925*y^38 + 276*y^37 - 62221*y^36 - 8388*y^35 + 868038*y^34 + 148164*y^33 - 8369604*y^32 - 1690536*y^31 + 57288192*y^30 + 13101552*y^29 - 281708427*y^28 - 70649728*y^27 + 997233012*y^26 + 267447468*y^25 - 2528669823*y^24 - 709905600*y^23 + 4548805662*y^22 + 1311105276*y^21 - 5738345406*y^20 - 1666490856*y^19 + 5023839240*y^18 + 1442807064*y^17 - 3028872840*y^16 - 846440256*y^15 + 1246931667*y^14 + 334448784*y^13 - 345422658*y^12 - 87695253*y^11 + 62698236*y^10 + 14793089*y^9 - 7127757*y^8 - 1516980*y^7 + 472108*y^6 + 85806*y^5 - 16338*y^4 - 2275*y^3 + 252*y^2 + 21*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 - 81*x^40 - 4*x^39 + 2925*x^38 + 276*x^37 - 62221*x^36 - 8388*x^35 + 868038*x^34 + 148164*x^33 - 8369604*x^32 - 1690536*x^31 + 57288192*x^30 + 13101552*x^29 - 281708427*x^28 - 70649728*x^27 + 997233012*x^26 + 267447468*x^25 - 2528669823*x^24 - 709905600*x^23 + 4548805662*x^22 + 1311105276*x^21 - 5738345406*x^20 - 1666490856*x^19 + 5023839240*x^18 + 1442807064*x^17 - 3028872840*x^16 - 846440256*x^15 + 1246931667*x^14 + 334448784*x^13 - 345422658*x^12 - 87695253*x^11 + 62698236*x^10 + 14793089*x^9 - 7127757*x^8 - 1516980*x^7 + 472108*x^6 + 85806*x^5 - 16338*x^4 - 2275*x^3 + 252*x^2 + 21*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]])