\\ Pari/GP code for working with number field 28.2.2657211910834657108824251865653514862060546875.1

\\ (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^28 - 9*x^27 + 53*x^26 - 226*x^25 + 695*x^24 - 1455*x^23 + 1352*x^22 + 4587*x^21 - 26437*x^20 + 75511*x^19 - 148243*x^18 + 166438*x^17 - 8850*x^16 - 553583*x^15 + 1599191*x^14 - 2514414*x^13 + 2904360*x^12 - 819447*x^11 - 1193138*x^10 + 5517125*x^9 - 17624126*x^8 + 7384001*x^7 - 17552991*x^6 + 11860646*x^5 - 8713168*x^4 - 7406098*x^3 - 9688274*x^2 - 1853761*x - 465059, 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]])