\\ Pari/GP code for working with number field 44.0.65347506006797164323533696798768413693852562329201668296707932160000000000000000000000.3 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^44 - 5*y^42 + 25*y^40 - 125*y^38 + 625*y^36 - 3125*y^34 + 15625*y^32 - 78125*y^30 + 390625*y^28 - 1953125*y^26 + 9765625*y^24 - 48828125*y^22 + 244140625*y^20 - 1220703125*y^18 + 6103515625*y^16 - 30517578125*y^14 + 152587890625*y^12 - 762939453125*y^10 + 3814697265625*y^8 - 19073486328125*y^6 + 95367431640625*y^4 - 476837158203125*y^2 + 2384185791015625, 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^44 - 5*x^42 + 25*x^40 - 125*x^38 + 625*x^36 - 3125*x^34 + 15625*x^32 - 78125*x^30 + 390625*x^28 - 1953125*x^26 + 9765625*x^24 - 48828125*x^22 + 244140625*x^20 - 1220703125*x^18 + 6103515625*x^16 - 30517578125*x^14 + 152587890625*x^12 - 762939453125*x^10 + 3814697265625*x^8 - 19073486328125*x^6 + 95367431640625*x^4 - 476837158203125*x^2 + 2384185791015625, 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]])