\\ Pari/GP code for working with number field 32.0.2553263220825544945190771147906377486172160000000000000000.2 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^32 - 2*y^31 - 39*y^30 + 78*y^29 + 678*y^28 - 1336*y^27 - 6933*y^26 + 13256*y^25 + 46161*y^24 - 84484*y^23 - 207785*y^22 + 363100*y^21 + 629290*y^20 - 1071214*y^19 - 1236459*y^18 + 2000398*y^17 + 1905251*y^16 - 877660*y^15 - 6200876*y^14 - 2748360*y^13 + 11687032*y^12 - 12604592*y^11 + 40542960*y^10 - 4541024*y^9 + 55107776*y^8 + 1846464*y^7 + 23026880*y^6 + 1271424*y^5 + 3919488*y^4 + 158976*y^3 + 231424*y^2 + 256, 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^32 - 2*x^31 - 39*x^30 + 78*x^29 + 678*x^28 - 1336*x^27 - 6933*x^26 + 13256*x^25 + 46161*x^24 - 84484*x^23 - 207785*x^22 + 363100*x^21 + 629290*x^20 - 1071214*x^19 - 1236459*x^18 + 2000398*x^17 + 1905251*x^16 - 877660*x^15 - 6200876*x^14 - 2748360*x^13 + 11687032*x^12 - 12604592*x^11 + 40542960*x^10 - 4541024*x^9 + 55107776*x^8 + 1846464*x^7 + 23026880*x^6 + 1271424*x^5 + 3919488*x^4 + 158976*x^3 + 231424*x^2 + 256, 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]])