\\ Pari/GP code for working with number field 35.35.42664569157776260194145291158312909027607801510075223187489424993547224224125412187641.1

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

\\ Define the number field: 
K = bnfinit(y^35 - 2*y^34 - 147*y^33 + 384*y^32 + 8999*y^31 - 28384*y^30 - 300934*y^29 + 1116603*y^28 + 6027458*y^27 - 26518095*y^26 - 73782007*y^25 + 404019590*y^24 + 521840181*y^23 - 4069268247*y^22 - 1503940067*y^21 + 27441438677*y^20 - 6471979421*y^19 - 124171354119*y^18 + 81524582969*y^17 + 375114725905*y^16 - 364687984188*y^15 - 747339096335*y^14 + 926308497451*y^13 + 961292881275*y^12 - 1451786049968*y^11 - 771321612374*y^10 + 1427632011474*y^9 + 365928432486*y^8 - 868346663001*y^7 - 97103750919*y^6 + 312022190475*y^5 + 17703737306*y^4 - 59996070990*y^3 - 4579250401*y^2 + 4824514261*y + 750695467, 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 - 2*x^34 - 147*x^33 + 384*x^32 + 8999*x^31 - 28384*x^30 - 300934*x^29 + 1116603*x^28 + 6027458*x^27 - 26518095*x^26 - 73782007*x^25 + 404019590*x^24 + 521840181*x^23 - 4069268247*x^22 - 1503940067*x^21 + 27441438677*x^20 - 6471979421*x^19 - 124171354119*x^18 + 81524582969*x^17 + 375114725905*x^16 - 364687984188*x^15 - 747339096335*x^14 + 926308497451*x^13 + 961292881275*x^12 - 1451786049968*x^11 - 771321612374*x^10 + 1427632011474*x^9 + 365928432486*x^8 - 868346663001*x^7 - 97103750919*x^6 + 312022190475*x^5 + 17703737306*x^4 - 59996070990*x^3 - 4579250401*x^2 + 4824514261*x + 750695467, 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]])