\\ Pari/GP code for working with number field 29.1.3587518960220469771354937124331329329937375710441.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^29 - y^28 - 7*y^27 - 50*y^26 + 119*y^25 + 495*y^24 + 59*y^23 - 530*y^22 + 637*y^21 + 2591*y^20 + 2977*y^19 + 1178*y^18 + 2213*y^17 + 5385*y^16 + 5965*y^15 + 6650*y^14 + 10883*y^13 + 16281*y^12 + 23955*y^11 + 21146*y^10 - 8059*y^9 - 7103*y^8 + 28957*y^7 + 37734*y^6 + 35524*y^5 + 2552*y^4 - 22800*y^3 - 19616*y^2 - 17024*y - 11776, 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^29 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776, 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]])