\\ Pari/GP code for working with number field 45.45.16527441490674381067348948889146522048718287580785548225593089356497792068001085725808918264262962961.1

\\ (Note that not all these functions may be available, and some may take a long time to execute.)

\\ Define the number field: 
K = bnfinit(x^45 - 4*x^44 - 114*x^43 + 468*x^42 + 5696*x^41 - 24132*x^40 - 165566*x^39 + 729869*x^38 + 3130045*x^37 - 14526130*x^36 - 40702810*x^35 + 202181480*x^34 + 374739344*x^33 - 2040701711*x^32 - 2468141099*x^31 + 15280860430*x^30 + 11523689449*x^29 - 86102308596*x^28 - 36476021474*x^27 + 368042106614*x^26 + 66266177672*x^25 - 1197228116258*x^24 + 511532340*x^23 + 2959992694385*x^22 - 394556499523*x^21 - 5529580093472*x^20 + 1265687493078*x^19 + 7719490702430*x^18 - 2271295840968*x^17 - 7914078560767*x^16 + 2645087732416*x^15 + 5804631677810*x^14 - 2048306537737*x^13 - 2929193071000*x^12 + 1039444940408*x^11 + 957147248584*x^10 - 333732759991*x^9 - 182597145321*x^8 + 64595300484*x^7 + 16257686900*x^6 - 6916885901*x^5 - 166352258*x^4 + 298954277*x^3 - 40977465*x^2 + 2118031*x - 36851, 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

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
p = 7;  \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
idealfactors = idealprimedec(K, p); \\ get the data
vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])