\\ Pari/GP code for working with number field 35.35.4974456597909264757129569420535951139863442014128922856612718940624179015847481515681.1

\\ 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 - 181*y^33 + 470*y^32 + 14259*y^31 - 45478*y^30 - 638530*y^29 + 2447153*y^28 + 17767052*y^27 - 82345759*y^26 - 312539633*y^25 + 1833247298*y^24 + 3264215879*y^23 - 27715265019*y^22 - 13487454777*y^21 + 286033632049*y^20 - 123264134993*y^19 - 1983204888009*y^18 + 2337315606897*y^17 + 8777927035267*y^16 - 17238219604018*y^15 - 21289728738165*y^14 + 71563023595709*y^13 + 8692436762607*y^12 - 169783075654432*y^11 + 94786315419610*y^10 + 202072200901306*y^9 - 240242794434974*y^8 - 58566297948415*y^7 + 212040077699383*y^6 - 73316281317651*y^5 - 51014137537536*y^4 + 37974482870362*y^3 - 1868368126575*y^2 - 3872034006535*y + 804494744591, 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 - 181*x^33 + 470*x^32 + 14259*x^31 - 45478*x^30 - 638530*x^29 + 2447153*x^28 + 17767052*x^27 - 82345759*x^26 - 312539633*x^25 + 1833247298*x^24 + 3264215879*x^23 - 27715265019*x^22 - 13487454777*x^21 + 286033632049*x^20 - 123264134993*x^19 - 1983204888009*x^18 + 2337315606897*x^17 + 8777927035267*x^16 - 17238219604018*x^15 - 21289728738165*x^14 + 71563023595709*x^13 + 8692436762607*x^12 - 169783075654432*x^11 + 94786315419610*x^10 + 202072200901306*x^9 - 240242794434974*x^8 - 58566297948415*x^7 + 212040077699383*x^6 - 73316281317651*x^5 - 51014137537536*x^4 + 37974482870362*x^3 - 1868368126575*x^2 - 3872034006535*x + 804494744591, 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]])