\\ Pari/GP code for working with number field 35.35.180307950435339664981987743016246180767325053322928227488471312870141585227228119921.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^35 - y^34 - 136*y^33 + 115*y^32 + 7922*y^31 - 5864*y^30 - 262379*y^29 + 176786*y^28 + 5536174*y^27 - 3463751*y^26 - 78912031*y^25 + 45605901*y^24 + 785104244*y^23 - 406720184*y^22 - 5548252572*y^21 + 2436396552*y^20 + 28031325744*y^19 - 9529852980*y^18 - 100888672963*y^17 + 22693210859*y^16 + 254967397769*y^15 - 26498170926*y^14 - 440501997963*y^13 - 3333796454*y^12 + 500538383949*y^11 + 44640325883*y^10 - 358942724140*y^9 - 43476115800*y^8 + 159203701176*y^7 + 16418359222*y^6 - 42718375268*y^5 - 1999390430*y^4 + 6413147106*y^3 - 204719982*y^2 - 415325205*y + 48529823, 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^35 - x^34 - 136*x^33 + 115*x^32 + 7922*x^31 - 5864*x^30 - 262379*x^29 + 176786*x^28 + 5536174*x^27 - 3463751*x^26 - 78912031*x^25 + 45605901*x^24 + 785104244*x^23 - 406720184*x^22 - 5548252572*x^21 + 2436396552*x^20 + 28031325744*x^19 - 9529852980*x^18 - 100888672963*x^17 + 22693210859*x^16 + 254967397769*x^15 - 26498170926*x^14 - 440501997963*x^13 - 3333796454*x^12 + 500538383949*x^11 + 44640325883*x^10 - 358942724140*x^9 - 43476115800*x^8 + 159203701176*x^7 + 16418359222*x^6 - 42718375268*x^5 - 1999390430*x^4 + 6413147106*x^3 - 204719982*x^2 - 415325205*x + 48529823, 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]])