\\ Pari/GP code for working with number field 27.1.1287062756823986424273208421285283554009333351.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^27 - 4*y^26 + 28*y^25 - 42*y^24 + 71*y^23 + 267*y^22 - 142*y^21 + 385*y^20 + 2210*y^19 + 2329*y^18 - 3315*y^17 + 15117*y^16 + 19014*y^15 - 23157*y^14 + 44479*y^13 + 77004*y^12 - 41903*y^11 + 19829*y^10 + 208952*y^9 - 35302*y^8 - 53453*y^7 + 258838*y^6 - 35701*y^5 + 60990*y^4 - 5393*y^3 + 76227*y^2 + 19031*y + 2209, 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^27 - 4*x^26 + 28*x^25 - 42*x^24 + 71*x^23 + 267*x^22 - 142*x^21 + 385*x^20 + 2210*x^19 + 2329*x^18 - 3315*x^17 + 15117*x^16 + 19014*x^15 - 23157*x^14 + 44479*x^13 + 77004*x^12 - 41903*x^11 + 19829*x^10 + 208952*x^9 - 35302*x^8 - 53453*x^7 + 258838*x^6 - 35701*x^5 + 60990*x^4 - 5393*x^3 + 76227*x^2 + 19031*x + 2209, 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]])