\\ Pari/GP code for working with number field 34.0.3325466068076643664357827857598159738994734276327509143073421552355865283.1

\\ 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 + 49*y^32 - 162*y^31 + 1646*y^30 - 6093*y^29 + 37764*y^28 - 140505*y^27 + 642029*y^26 - 2158868*y^25 + 7867204*y^24 - 23451724*y^23 + 70989641*y^22 - 182710441*y^21 + 459461291*y^20 - 1006591547*y^19 + 2135502310*y^18 - 4017190895*y^17 + 7274499825*y^16 - 11714663796*y^15 + 18043531688*y^14 - 24702818067*y^13 + 32221074210*y^12 - 36915149197*y^11 + 39648896356*y^10 - 36487346558*y^9 + 31372407145*y^8 - 23064375767*y^7 + 16052201854*y^6 - 9172737675*y^5 + 4518648355*y^4 - 1618857168*y^3 + 448908855*y^2 - 73089631*y + 8048569, 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 + 49*x^32 - 162*x^31 + 1646*x^30 - 6093*x^29 + 37764*x^28 - 140505*x^27 + 642029*x^26 - 2158868*x^25 + 7867204*x^24 - 23451724*x^23 + 70989641*x^22 - 182710441*x^21 + 459461291*x^20 - 1006591547*x^19 + 2135502310*x^18 - 4017190895*x^17 + 7274499825*x^16 - 11714663796*x^15 + 18043531688*x^14 - 24702818067*x^13 + 32221074210*x^12 - 36915149197*x^11 + 39648896356*x^10 - 36487346558*x^9 + 31372407145*x^8 - 23064375767*x^7 + 16052201854*x^6 - 9172737675*x^5 + 4518648355*x^4 - 1618857168*x^3 + 448908855*x^2 - 73089631*x + 8048569, 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]])