\\ Pari/GP code for working with number field 35.35.42036332355782985928056054963938846044720164955104674785517585227702767658858707400475847187116419041.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^35 - y^34 - 442*y^33 + 217*y^32 + 85220*y^31 - 2238*y^30 - 9436469*y^29 - 3733012*y^28 + 665405852*y^27 + 518480639*y^26 - 31337617705*y^25 - 35516568513*y^24 + 1006061093515*y^23 + 1481171485580*y^22 - 22064557661077*y^21 - 40105273516115*y^20 + 325564905075174*y^19 + 718606613070565*y^18 - 3104378497697910*y^17 - 8465726517162874*y^16 + 17359496453493293*y^15 + 63647428757599053*y^14 - 39918422702860182*y^13 - 286218584896036781*y^12 - 91666282635987596*y^11 + 665845380121245570*y^10 + 709902006827264737*y^9 - 485410723276588718*y^8 - 1192541413062255796*y^7 - 419664407183170897*y^6 + 445121936202637191*y^5 + 400491751414674915*y^4 + 50965550537459510*y^3 - 54268060119801121*y^2 - 22150406599986335*y - 2362460740247707, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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 - x^34 - 442*x^33 + 217*x^32 + 85220*x^31 - 2238*x^30 - 9436469*x^29 - 3733012*x^28 + 665405852*x^27 + 518480639*x^26 - 31337617705*x^25 - 35516568513*x^24 + 1006061093515*x^23 + 1481171485580*x^22 - 22064557661077*x^21 - 40105273516115*x^20 + 325564905075174*x^19 + 718606613070565*x^18 - 3104378497697910*x^17 - 8465726517162874*x^16 + 17359496453493293*x^15 + 63647428757599053*x^14 - 39918422702860182*x^13 - 286218584896036781*x^12 - 91666282635987596*x^11 + 665845380121245570*x^10 + 709902006827264737*x^9 - 485410723276588718*x^8 - 1192541413062255796*x^7 - 419664407183170897*x^6 + 445121936202637191*x^5 + 400491751414674915*x^4 + 50965550537459510*x^3 - 54268060119801121*x^2 - 22150406599986335*x - 2362460740247707, 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(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])