\\ Pari/GP code for working with number field 45.45.14871644565877379464047221413905669257566616785997555307108986009149096115256167660961155530887613343502969.1 \\ Some of these functions may take a long time to execute (this depends on the field). \\ Define the number field: K = bnfinit(y^45 - 9*y^44 - 117*y^43 + 1356*y^42 + 4455*y^41 - 86382*y^40 + 4596*y^39 + 3024342*y^38 - 5886234*y^37 - 62776190*y^36 + 223064037*y^35 + 751420521*y^34 - 4412144961*y^33 - 3725440704*y^32 + 54057041307*y^31 - 28786222110*y^30 - 426126343077*y^29 + 673008341841*y^28 + 2086637912508*y^27 - 5873721786378*y^26 - 5053254569676*y^25 + 30385493847345*y^24 - 5904503198856*y^23 - 98922911866245*y^22 + 93605350409913*y^21 + 191046717571311*y^20 - 349809055484526*y^19 - 147155959283342*y^18 + 707251464652203*y^17 - 217826629489476*y^16 - 797405893685064*y^15 + 705978096992721*y^14 + 378713501361567*y^13 - 782478874269585*y^12 + 139525392055671*y^11 + 385252617992454*y^10 - 250562776470715*y^9 - 36607492213488*y^8 + 95351499735039*y^7 - 31186566059442*y^6 - 5609742817023*y^5 + 6576945066567*y^4 - 1857401457582*y^3 + 224711418717*y^2 - 8430571116*y - 155181097, 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^45 - 9*x^44 - 117*x^43 + 1356*x^42 + 4455*x^41 - 86382*x^40 + 4596*x^39 + 3024342*x^38 - 5886234*x^37 - 62776190*x^36 + 223064037*x^35 + 751420521*x^34 - 4412144961*x^33 - 3725440704*x^32 + 54057041307*x^31 - 28786222110*x^30 - 426126343077*x^29 + 673008341841*x^28 + 2086637912508*x^27 - 5873721786378*x^26 - 5053254569676*x^25 + 30385493847345*x^24 - 5904503198856*x^23 - 98922911866245*x^22 + 93605350409913*x^21 + 191046717571311*x^20 - 349809055484526*x^19 - 147155959283342*x^18 + 707251464652203*x^17 - 217826629489476*x^16 - 797405893685064*x^15 + 705978096992721*x^14 + 378713501361567*x^13 - 782478874269585*x^12 + 139525392055671*x^11 + 385252617992454*x^10 - 250562776470715*x^9 - 36607492213488*x^8 + 95351499735039*x^7 - 31186566059442*x^6 - 5609742817023*x^5 + 6576945066567*x^4 - 1857401457582*x^3 + 224711418717*x^2 - 8430571116*x - 155181097, 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]])