\\ Pari/GP code for working with number field 26.0.1999192748751319434913358682729045110107421875.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^26 - 6*y^25 + 34*y^24 - 149*y^23 + 724*y^22 - 3027*y^21 + 10234*y^20 - 26698*y^19 + 56427*y^18 - 101122*y^17 + 164719*y^16 - 254230*y^15 + 377196*y^14 - 535703*y^13 + 716143*y^12 - 954928*y^11 + 1568217*y^10 - 3148667*y^9 + 5745440*y^8 - 8171380*y^7 + 9122775*y^6 - 8433000*y^5 + 7096875*y^4 - 5099875*y^3 + 3012500*y^2 - 717500*y + 153125, 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 - 6*x^25 + 34*x^24 - 149*x^23 + 724*x^22 - 3027*x^21 + 10234*x^20 - 26698*x^19 + 56427*x^18 - 101122*x^17 + 164719*x^16 - 254230*x^15 + 377196*x^14 - 535703*x^13 + 716143*x^12 - 954928*x^11 + 1568217*x^10 - 3148667*x^9 + 5745440*x^8 - 8171380*x^7 + 9122775*x^6 - 8433000*x^5 + 7096875*x^4 - 5099875*x^3 + 3012500*x^2 - 717500*x + 153125, 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]])