\\ Pari/GP code for working with number field 42.42.98118980687896783910098639727548084722605054105289047332129555342401833439729.1

\\ 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 - 42*y^40 + 42*y^39 + 818*y^38 - 818*y^37 - 9803*y^36 + 9803*y^35 + 80884*y^34 - 80884*y^33 - 487103*y^32 + 487103*y^31 + 2214673*y^30 - 2214673*y^29 - 7756167*y^28 + 7756167*y^27 + 21159269*y^26 - 21159269*y^25 - 45176143*y^24 + 45176143*y^23 + 75433697*y^22 - 75433697*y^21 - 97942948*y^20 + 97942948*y^19 + 97804877*y^18 - 97804877*y^17 - 73850908*y^16 + 73850908*y^15 + 41150012*y^14 - 41150012*y^13 - 16350448*y^12 + 16350448*y^11 + 4413607*y^10 - 4413607*y^9 - 753918*y^8 + 753918*y^7 + 72886*y^6 - 72886*y^5 - 3267*y^4 + 3267*y^3 + 44*y^2 - 44*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 - 42*x^40 + 42*x^39 + 818*x^38 - 818*x^37 - 9803*x^36 + 9803*x^35 + 80884*x^34 - 80884*x^33 - 487103*x^32 + 487103*x^31 + 2214673*x^30 - 2214673*x^29 - 7756167*x^28 + 7756167*x^27 + 21159269*x^26 - 21159269*x^25 - 45176143*x^24 + 45176143*x^23 + 75433697*x^22 - 75433697*x^21 - 97942948*x^20 + 97942948*x^19 + 97804877*x^18 - 97804877*x^17 - 73850908*x^16 + 73850908*x^15 + 41150012*x^14 - 41150012*x^13 - 16350448*x^12 + 16350448*x^11 + 4413607*x^10 - 4413607*x^9 - 753918*x^8 + 753918*x^7 + 72886*x^6 - 72886*x^5 - 3267*x^4 + 3267*x^3 + 44*x^2 - 44*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]])