\\ Pari/GP code for working with number field 35.35.705021958507555897735769309192822159832506785420715156309512394727789796888828277587890625.1

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

\\ Define the number field: 
K = bnfinit(y^35 - 175*y^33 - 70*y^32 + 13125*y^31 + 9702*y^30 - 556640*y^29 - 573225*y^28 + 14844900*y^27 + 19050185*y^26 - 261854635*y^25 - 394847250*y^24 + 3128435905*y^23 + 5347852195*y^22 - 25515383355*y^21 - 48297678121*y^20 + 141795944465*y^19 + 292585456625*y^18 - 533394487115*y^17 - 1187150912745*y^16 + 1349113537702*y^15 + 3207931617015*y^14 - 2286376450785*y^13 - 5710072444965*y^12 + 2606036871270*y^11 + 6560227221870*y^10 - 2026242763720*y^9 - 4691478893710*y^8 + 1084067118185*y^7 + 1962115542205*y^6 - 373976828869*y^5 - 432767079290*y^4 + 65717425060*y^3 + 43746645075*y^2 - 3865111915*y - 1645272349, 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 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349, 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]])