\\ Pari/GP code for working with number field 25.5.722359806654877741268938522624.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^25 - 3*y^24 + 4*y^23 - 7*y^22 + 5*y^21 - 6*y^20 + 28*y^19 - 37*y^18 + 24*y^17 + 60*y^15 - 133*y^14 + 74*y^13 + 37*y^12 - 24*y^11 - 112*y^10 + 155*y^9 - 56*y^8 - 45*y^7 + 51*y^6 - 17*y^5 + 3*y^4 - 3*y^3 + 3*y - 1, 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^25 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1, 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]])