\\ Pari/GP code for working with number field 40.0.2048466217933115502043842255668249817786415413279145255704960577548561.1

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

\\ Define the number field: 
K = bnfinit(y^40 - y^39 - 3*y^38 + 10*y^37 - 10*y^36 - 30*y^35 + 127*y^34 - 127*y^33 - 381*y^32 + 1540*y^31 - 1540*y^30 + 5017*y^29 + 9192*y^28 - 47740*y^27 + 39883*y^26 + 133500*y^25 - 518980*y^24 + 534289*y^23 + 1583184*y^22 - 6478780*y^21 + 6419731*y^20 + 19436340*y^19 + 14248656*y^18 - 14425803*y^17 - 42037380*y^16 - 32440500*y^15 + 29074707*y^14 + 104407380*y^13 + 60308712*y^12 - 98749611*y^11 - 90935460*y^10 - 272806380*y^9 - 202479021*y^8 + 202479021*y^7 + 607437063*y^6 + 430467210*y^5 - 430467210*y^4 - 1291401630*y^3 - 1162261467*y^2 + 1162261467*y + 3486784401, 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^40 - x^39 - 3*x^38 + 10*x^37 - 10*x^36 - 30*x^35 + 127*x^34 - 127*x^33 - 381*x^32 + 1540*x^31 - 1540*x^30 + 5017*x^29 + 9192*x^28 - 47740*x^27 + 39883*x^26 + 133500*x^25 - 518980*x^24 + 534289*x^23 + 1583184*x^22 - 6478780*x^21 + 6419731*x^20 + 19436340*x^19 + 14248656*x^18 - 14425803*x^17 - 42037380*x^16 - 32440500*x^15 + 29074707*x^14 + 104407380*x^13 + 60308712*x^12 - 98749611*x^11 - 90935460*x^10 - 272806380*x^9 - 202479021*x^8 + 202479021*x^7 + 607437063*x^6 + 430467210*x^5 - 430467210*x^4 - 1291401630*x^3 - 1162261467*x^2 + 1162261467*x + 3486784401, 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]])