\\ Pari/GP code for working with number field 28.0.14121388821225670988853483488774192350843817726481.1

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

\\ Define the number field: 
K = bnfinit(y^28 - y^27 - y^26 - 50*y^25 + 45*y^24 + 41*y^23 + 923*y^22 - 740*y^21 - 566*y^20 - 7986*y^19 + 5482*y^18 + 3755*y^17 + 34135*y^16 - 15421*y^15 - 15266*y^14 - 71623*y^13 - 3964*y^12 + 45122*y^11 + 87041*y^10 + 49398*y^9 - 13057*y^8 - 105029*y^7 - 52559*y^6 - 58937*y^5 + 57132*y^4 + 53384*y^3 + 9659*y^2 + 13272*y + 6241, 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^28 - x^27 - x^26 - 50*x^25 + 45*x^24 + 41*x^23 + 923*x^22 - 740*x^21 - 566*x^20 - 7986*x^19 + 5482*x^18 + 3755*x^17 + 34135*x^16 - 15421*x^15 - 15266*x^14 - 71623*x^13 - 3964*x^12 + 45122*x^11 + 87041*x^10 + 49398*x^9 - 13057*x^8 - 105029*x^7 - 52559*x^6 - 58937*x^5 + 57132*x^4 + 53384*x^3 + 9659*x^2 + 13272*x + 6241, 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]])