\\ Pari/GP code for working with number field 45.45.2524373658125154706394844918628378076658288807428864227492811533944564375098081397291186234112097170769516929.1

\\ Some of these functions may take a long time to execute (this depends on the field).

\\ Define the number field: 
K = bnfinit(y^45 - 12*y^44 - 80*y^43 + 1424*y^42 + 1516*y^41 - 74554*y^40 + 63158*y^39 + 2282501*y^38 - 4255902*y^37 - 45636421*y^36 + 116693559*y^35 + 630121996*y^34 - 1971613347*y^33 - 6193698077*y^32 + 22855343401*y^31 + 44007670585*y^30 - 190617019593*y^29 - 226920195907*y^28 + 1172746429282*y^27 + 842990049201*y^26 - 5394671280584*y^25 - 2206384597765*y^24 + 18666651459717*y^23 + 3871477687999*y^22 - 48596558981735*y^21 - 4036750693451*y^20 + 94693820108958*y^19 + 1546861746446*y^18 - 136570614884744*y^17 + 956863921562*y^16 + 143064103358001*y^15 + 102341290612*y^14 - 105649963477592*y^13 - 2816833651750*y^12 + 52484763220205*y^11 + 2919265367283*y^10 - 16311452643376*y^9 - 1104335997267*y^8 + 2857645920655*y^7 + 105919440856*y^6 - 255094574859*y^5 + 8466426668*y^4 + 8654209646*y^3 - 1026667213*y^2 + 14388119*y + 1507921, 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 - 12*x^44 - 80*x^43 + 1424*x^42 + 1516*x^41 - 74554*x^40 + 63158*x^39 + 2282501*x^38 - 4255902*x^37 - 45636421*x^36 + 116693559*x^35 + 630121996*x^34 - 1971613347*x^33 - 6193698077*x^32 + 22855343401*x^31 + 44007670585*x^30 - 190617019593*x^29 - 226920195907*x^28 + 1172746429282*x^27 + 842990049201*x^26 - 5394671280584*x^25 - 2206384597765*x^24 + 18666651459717*x^23 + 3871477687999*x^22 - 48596558981735*x^21 - 4036750693451*x^20 + 94693820108958*x^19 + 1546861746446*x^18 - 136570614884744*x^17 + 956863921562*x^16 + 143064103358001*x^15 + 102341290612*x^14 - 105649963477592*x^13 - 2816833651750*x^12 + 52484763220205*x^11 + 2919265367283*x^10 - 16311452643376*x^9 - 1104335997267*x^8 + 2857645920655*x^7 + 105919440856*x^6 - 255094574859*x^5 + 8466426668*x^4 + 8654209646*x^3 - 1026667213*x^2 + 14388119*x + 1507921, 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]])