\\ Pari/GP code for working with number field 42.0.4472772095648327544266441640449519840379754819764800031247757188817501068115234375.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^42 - y^41 + 44*y^40 - 44*y^39 + 904*y^38 - 904*y^37 + 11525*y^36 - 11525*y^35 + 102212*y^34 - 102212*y^33 + 670199*y^32 - 670199*y^31 + 3371975*y^30 - 3371975*y^29 + 13342815*y^28 - 13342815*y^27 + 42258251*y^26 - 42258251*y^25 + 108593663*y^24 - 108593663*y^23 + 229203503*y^22 - 229203503*y^21 + 402580148*y^20 - 402580148*y^19 + 598327973*y^18 - 598327973*y^17 + 769983758*y^16 - 769983758*y^15 + 884984678*y^14 - 884984678*y^13 + 942485138*y^12 - 942485138*y^11 + 963249193*y^10 - 963249193*y^9 + 968416718*y^8 - 968416718*y^7 + 969243522*y^6 - 969243522*y^5 + 969319675*y^4 - 969319675*y^3 + 969322986*y^2 - 969322986*y + 969323029, 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^42 - x^41 + 44*x^40 - 44*x^39 + 904*x^38 - 904*x^37 + 11525*x^36 - 11525*x^35 + 102212*x^34 - 102212*x^33 + 670199*x^32 - 670199*x^31 + 3371975*x^30 - 3371975*x^29 + 13342815*x^28 - 13342815*x^27 + 42258251*x^26 - 42258251*x^25 + 108593663*x^24 - 108593663*x^23 + 229203503*x^22 - 229203503*x^21 + 402580148*x^20 - 402580148*x^19 + 598327973*x^18 - 598327973*x^17 + 769983758*x^16 - 769983758*x^15 + 884984678*x^14 - 884984678*x^13 + 942485138*x^12 - 942485138*x^11 + 963249193*x^10 - 963249193*x^9 + 968416718*x^8 - 968416718*x^7 + 969243522*x^6 - 969243522*x^5 + 969319675*x^4 - 969319675*x^3 + 969322986*x^2 - 969322986*x + 969323029, 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]])