\\ Pari/GP code for working with number field 34.34.342523005011894297428856269332610453116457630461733441736562419892654124149.1

\\ 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 - 101*y^32 - 39*y^31 + 4475*y^30 + 7317*y^29 - 107886*y^28 - 308922*y^27 + 1428035*y^26 + 6473545*y^25 - 8038724*y^24 - 76649821*y^23 - 34999669*y^22 + 511384499*y^21 + 872682200*y^20 - 1610027003*y^19 - 5837837426*y^18 - 597495179*y^17 + 18112177824*y^16 + 20594193249*y^15 - 20555590855*y^14 - 57678181587*y^13 - 18084925848*y^12 + 57474448334*y^11 + 58215655753*y^10 - 9898232087*y^9 - 42024711035*y^8 - 14514531077*y^7 + 10323419848*y^6 + 7567081746*y^5 - 122533277*y^4 - 1123034742*y^3 - 160716846*y^2 + 51273995*y + 10497967, 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 - 101*x^32 - 39*x^31 + 4475*x^30 + 7317*x^29 - 107886*x^28 - 308922*x^27 + 1428035*x^26 + 6473545*x^25 - 8038724*x^24 - 76649821*x^23 - 34999669*x^22 + 511384499*x^21 + 872682200*x^20 - 1610027003*x^19 - 5837837426*x^18 - 597495179*x^17 + 18112177824*x^16 + 20594193249*x^15 - 20555590855*x^14 - 57678181587*x^13 - 18084925848*x^12 + 57474448334*x^11 + 58215655753*x^10 - 9898232087*x^9 - 42024711035*x^8 - 14514531077*x^7 + 10323419848*x^6 + 7567081746*x^5 - 122533277*x^4 - 1123034742*x^3 - 160716846*x^2 + 51273995*x + 10497967, 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]])