\\ Pari/GP code for working with number field 42.42.1236219045653317330764639083558080790009887216550178193020957667462329030021.1

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

\\ Define the number field: 
K = bnfinit(y^42 - 42*y^40 + 819*y^38 - 9842*y^36 - y^35 + 81585*y^34 + 35*y^33 - 494802*y^32 - 560*y^31 + 2272424*y^30 + 5425*y^29 - 8069424*y^28 - 35525*y^27 + 22428252*y^26 + 166257*y^25 - 49085400*y^24 - 573300*y^23 + 84672315*y^22 + 1480051*y^21 - 114717330*y^20 - 2877896*y^19 + 121090515*y^18 + 4206314*y^17 - 98285670*y^16 - 4577216*y^15 + 60174899*y^14 + 3643150*y^13 - 27041546*y^12 - 2063243*y^11 + 8580418*y^10 + 798357*y^9 - 1816836*y^8 - 198948*y^7 + 235249*y^6 + 29211*y^5 - 15974*y^4 - 2170*y^3 + 392*y^2 + 56*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 - 42*x^40 + 819*x^38 - 9842*x^36 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*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]])