\\ Pari/GP code for working with number field 38.0.566307980489842712804141765373005041569726061524232428759764421442954360220603558656123885789904896.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^38 + 229*y^36 + 22442*y^34 + 1242325*y^32 + 43174744*y^30 + 989928070*y^28 + 15310092242*y^26 + 160462320009*y^24 + 1129846290713*y^22 + 5241767939799*y^20 + 15619544332617*y^18 + 29338707057674*y^16 + 34962128493052*y^14 + 26729901012072*y^12 + 13161379227544*y^10 + 4139430890096*y^8 + 811133080337*y^6 + 94183964323*y^4 + 5864318791*y^2 + 149876149, 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^38 + 229*x^36 + 22442*x^34 + 1242325*x^32 + 43174744*x^30 + 989928070*x^28 + 15310092242*x^26 + 160462320009*x^24 + 1129846290713*x^22 + 5241767939799*x^20 + 15619544332617*x^18 + 29338707057674*x^16 + 34962128493052*x^14 + 26729901012072*x^12 + 13161379227544*x^10 + 4139430890096*x^8 + 811133080337*x^6 + 94183964323*x^4 + 5864318791*x^2 + 149876149, 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]])