\\ Pari/GP code for working with number field 26.2.23347285006439981249999555372409414171948497.1

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

\\ Define the number field: 
K = bnfinit(y^26 - y^25 - 50*y^24 + 48*y^23 + 1188*y^22 - 1066*y^21 - 17960*y^20 + 14231*y^19 + 194023*y^18 - 126696*y^17 - 1579415*y^16 + 794998*y^15 + 9883811*y^14 - 3670079*y^13 - 47524550*y^12 + 13070104*y^11 + 173418938*y^10 - 37145018*y^9 - 469905596*y^8 + 84248201*y^7 + 912345834*y^6 - 152992905*y^5 - 1188256605*y^4 + 203114817*y^3 + 913828487*y^2 - 132782216*y - 329129521, 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^26 - x^25 - 50*x^24 + 48*x^23 + 1188*x^22 - 1066*x^21 - 17960*x^20 + 14231*x^19 + 194023*x^18 - 126696*x^17 - 1579415*x^16 + 794998*x^15 + 9883811*x^14 - 3670079*x^13 - 47524550*x^12 + 13070104*x^11 + 173418938*x^10 - 37145018*x^9 - 469905596*x^8 + 84248201*x^7 + 912345834*x^6 - 152992905*x^5 - 1188256605*x^4 + 203114817*x^3 + 913828487*x^2 - 132782216*x - 329129521, 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]])