\\ Pari/GP code for working with number field 34.0.2652335238355663972863781109929452800183143879582476922663961790823.1

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

\\ Define the number field: 
K = bnfinit(y^34 - y^33 + 2*y^32 + 64*y^31 - 57*y^30 + 107*y^29 + 1603*y^28 - 1261*y^27 + 2206*y^26 + 20286*y^25 - 13891*y^24 + 22349*y^23 + 140429*y^22 - 82189*y^21 + 120996*y^20 + 538574*y^19 - 264119*y^18 + 327327*y^17 + 1143572*y^16 - 499882*y^15 + 128724*y^14 + 1425880*y^13 - 737516*y^12 - 1133952*y^11 + 1294431*y^10 - 1061173*y^9 - 1305757*y^8 + 907693*y^7 - 16051*y^6 - 713165*y^5 + 590348*y^4 + 474529*y^3 + 129602*y^2 - 93135*y + 56857, 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^34 - x^33 + 2*x^32 + 64*x^31 - 57*x^30 + 107*x^29 + 1603*x^28 - 1261*x^27 + 2206*x^26 + 20286*x^25 - 13891*x^24 + 22349*x^23 + 140429*x^22 - 82189*x^21 + 120996*x^20 + 538574*x^19 - 264119*x^18 + 327327*x^17 + 1143572*x^16 - 499882*x^15 + 128724*x^14 + 1425880*x^13 - 737516*x^12 - 1133952*x^11 + 1294431*x^10 - 1061173*x^9 - 1305757*x^8 + 907693*x^7 - 16051*x^6 - 713165*x^5 + 590348*x^4 + 474529*x^3 + 129602*x^2 - 93135*x + 56857, 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]])