\\ Pari/GP code for working with number field 26.2.14465938847694062481283347921338966064453125.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^26 - 3*y^25 + 8*y^24 - 23*y^23 - 44*y^22 + 17*y^21 + 450*y^20 - 860*y^19 + 3401*y^18 + 4044*y^17 - 12200*y^16 - 9996*y^15 + 5376*y^14 - 176112*y^13 - 67837*y^12 - 1501*y^11 - 261354*y^10 - 143521*y^9 - 1171*y^8 - 152170*y^7 - 30780*y^6 - 39875*y^5 - 16350*y^4 - 26625*y^3 - 13000*y^2 - 6875*y - 3125, 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^26 - 3*x^25 + 8*x^24 - 23*x^23 - 44*x^22 + 17*x^21 + 450*x^20 - 860*x^19 + 3401*x^18 + 4044*x^17 - 12200*x^16 - 9996*x^15 + 5376*x^14 - 176112*x^13 - 67837*x^12 - 1501*x^11 - 261354*x^10 - 143521*x^9 - 1171*x^8 - 152170*x^7 - 30780*x^6 - 39875*x^5 - 16350*x^4 - 26625*x^3 - 13000*x^2 - 6875*x - 3125, 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]])