\\ Pari/GP code for working with number field 35.35.103338030412840513192336580932106187652481378569380154885948286391794681549072265625.1

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

\\ Define the number field: 
K = bnfinit(y^35 - 5*y^34 - 120*y^33 + 560*y^32 + 6265*y^31 - 27063*y^30 - 188895*y^29 + 746800*y^28 + 3677845*y^27 - 13130310*y^26 - 48923530*y^25 + 155517810*y^24 + 458129250*y^23 - 1279983515*y^22 - 3063191465*y^21 + 7446901792*y^20 + 14674072475*y^19 - 30859715040*y^18 - 50096318770*y^17 + 91037787910*y^16 + 120155286235*y^15 - 189447334860*y^14 - 197298653000*y^13 + 272612081065*y^12 + 212356717105*y^11 - 261877354307*y^10 - 139209319745*y^9 + 158110983885*y^8 + 48766657725*y^7 - 53789658890*y^6 - 7150843552*y^5 + 8233276160*y^4 + 456884545*y^3 - 363355535*y^2 - 50594420*y - 1782107, 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^35 - 5*x^34 - 120*x^33 + 560*x^32 + 6265*x^31 - 27063*x^30 - 188895*x^29 + 746800*x^28 + 3677845*x^27 - 13130310*x^26 - 48923530*x^25 + 155517810*x^24 + 458129250*x^23 - 1279983515*x^22 - 3063191465*x^21 + 7446901792*x^20 + 14674072475*x^19 - 30859715040*x^18 - 50096318770*x^17 + 91037787910*x^16 + 120155286235*x^15 - 189447334860*x^14 - 197298653000*x^13 + 272612081065*x^12 + 212356717105*x^11 - 261877354307*x^10 - 139209319745*x^9 + 158110983885*x^8 + 48766657725*x^7 - 53789658890*x^6 - 7150843552*x^5 + 8233276160*x^4 + 456884545*x^3 - 363355535*x^2 - 50594420*x - 1782107, 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]])