\\ Pari/GP code for working with number field 26.0.384766437057818380952237905666104641217782272563.1

\\ 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 + 25*y^24 - 14*y^23 + 405*y^22 - 192*y^21 + 3603*y^20 - 1111*y^19 + 22650*y^18 - 5414*y^17 + 87624*y^16 - 5096*y^15 + 234451*y^14 - 19756*y^13 + 367484*y^12 - 30562*y^11 + 404986*y^10 - 57146*y^9 + 232119*y^8 - 34742*y^7 + 91429*y^6 - 16226*y^5 + 7319*y^4 + 98*y^3 + 168*y^2 - 10*y + 1, 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 + 25*x^24 - 14*x^23 + 405*x^22 - 192*x^21 + 3603*x^20 - 1111*x^19 + 22650*x^18 - 5414*x^17 + 87624*x^16 - 5096*x^15 + 234451*x^14 - 19756*x^13 + 367484*x^12 - 30562*x^11 + 404986*x^10 - 57146*x^9 + 232119*x^8 - 34742*x^7 + 91429*x^6 - 16226*x^5 + 7319*x^4 + 98*x^3 + 168*x^2 - 10*x + 1, 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]])