\\ Pari/GP code for working with number field 28.2.295815184798509371659078776791937755272938949172963.1

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

\\ Define the number field: 
K = bnfinit(y^28 - 14*y^27 + 80*y^26 - 221*y^25 + 397*y^24 - 1904*y^23 + 10213*y^22 - 26576*y^21 + 32909*y^20 - 65079*y^19 + 350623*y^18 - 1046531*y^17 + 1678487*y^16 - 1771438*y^15 + 2670493*y^14 - 7195176*y^13 + 14367013*y^12 - 15585945*y^11 + 7205726*y^10 - 3208396*y^9 + 37523327*y^8 - 120026000*y^7 + 201372903*y^6 - 215912334*y^5 + 156455683*y^4 - 76691304*y^3 + 25200801*y^2 - 5337738*y + 544563, 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 - 14*x^27 + 80*x^26 - 221*x^25 + 397*x^24 - 1904*x^23 + 10213*x^22 - 26576*x^21 + 32909*x^20 - 65079*x^19 + 350623*x^18 - 1046531*x^17 + 1678487*x^16 - 1771438*x^15 + 2670493*x^14 - 7195176*x^13 + 14367013*x^12 - 15585945*x^11 + 7205726*x^10 - 3208396*x^9 + 37523327*x^8 - 120026000*x^7 + 201372903*x^6 - 215912334*x^5 + 156455683*x^4 - 76691304*x^3 + 25200801*x^2 - 5337738*x + 544563, 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]])