\\ Pari/GP code for working with number field 25.1.11843998797823466973474682749162211402125799921.1

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

\\ Define the number field: 
K = bnfinit(y^25 - y^24 - 19*y^23 - 76*y^22 + 156*y^21 + 1005*y^20 + 1755*y^19 - 1564*y^18 - 17480*y^17 + 14003*y^16 - 40555*y^15 + 158147*y^14 - 336486*y^13 - 456783*y^12 + 2807964*y^11 - 9825280*y^10 + 32382901*y^9 - 51222985*y^8 + 76250162*y^7 - 82760080*y^6 + 46691010*y^5 + 4604688*y^4 - 3014982*y^3 - 12748972*y^2 + 5156073*y + 1384173, 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^25 - x^24 - 19*x^23 - 76*x^22 + 156*x^21 + 1005*x^20 + 1755*x^19 - 1564*x^18 - 17480*x^17 + 14003*x^16 - 40555*x^15 + 158147*x^14 - 336486*x^13 - 456783*x^12 + 2807964*x^11 - 9825280*x^10 + 32382901*x^9 - 51222985*x^8 + 76250162*x^7 - 82760080*x^6 + 46691010*x^5 + 4604688*x^4 - 3014982*x^3 - 12748972*x^2 + 5156073*x + 1384173, 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]])