\\ Pari/GP code for working with number field 26.2.33169014017307639762747533311185137385697693.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^26 - y^25 - 45*y^24 + 44*y^23 + 912*y^22 - 836*y^21 - 11055*y^20 + 9149*y^19 + 89968*y^18 - 63863*y^17 - 520434*y^16 + 297998*y^15 + 2198217*y^14 - 952653*y^13 - 6808591*y^12 + 2140242*y^11 + 15236220*y^10 - 3364016*y^9 - 23680109*y^8 + 2956997*y^7 + 23956149*y^6 - 618594*y^5 - 14015975*y^4 - 2419867*y^3 + 3438259*y^2 + 3561247*y - 1990843, 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 - x^25 - 45*x^24 + 44*x^23 + 912*x^22 - 836*x^21 - 11055*x^20 + 9149*x^19 + 89968*x^18 - 63863*x^17 - 520434*x^16 + 297998*x^15 + 2198217*x^14 - 952653*x^13 - 6808591*x^12 + 2140242*x^11 + 15236220*x^10 - 3364016*x^9 - 23680109*x^8 + 2956997*x^7 + 23956149*x^6 - 618594*x^5 - 14015975*x^4 - 2419867*x^3 + 3438259*x^2 + 3561247*x - 1990843, 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]])