\\ Pari/GP code for working with number field 42.0.959396304051793463814262846982490027578741814649477038563926538598268329263104.1

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

\\ Define the number field: 
K = bnfinit(y^42 + 41*y^40 + 780*y^38 + 9139*y^36 + 73815*y^34 + 435897*y^32 + 1947792*y^30 + 6724520*y^28 + 18156204*y^26 + 38567100*y^24 + 64512240*y^22 + 84672315*y^20 + 86493225*y^18 + 67863915*y^16 + 40116600*y^14 + 17383860*y^12 + 5311735*y^10 + 1081575*y^8 + 134596*y^6 + 8855*y^4 + 231*y^2 + 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 + 41*x^40 + 780*x^38 + 9139*x^36 + 73815*x^34 + 435897*x^32 + 1947792*x^30 + 6724520*x^28 + 18156204*x^26 + 38567100*x^24 + 64512240*x^22 + 84672315*x^20 + 86493225*x^18 + 67863915*x^16 + 40116600*x^14 + 17383860*x^12 + 5311735*x^10 + 1081575*x^8 + 134596*x^6 + 8855*x^4 + 231*x^2 + 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]])