\\ Pari/GP code for working with number field 26.2.713943735055873357083710772983186869616893.1

\\ 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 - 37*y^24 + 36*y^23 + 656*y^22 - 602*y^21 - 7532*y^20 + 5987*y^19 + 63242*y^18 - 39168*y^17 - 407978*y^16 + 177750*y^15 + 2044391*y^14 - 593805*y^13 - 7863783*y^12 + 1620832*y^11 + 22802242*y^10 - 3804296*y^9 - 48757198*y^8 + 7281003*y^7 + 74301119*y^6 - 12442481*y^5 - 74591007*y^4 + 16397553*y^3 + 41639217*y^2 - 9412671*y - 11716513, 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 - 37*x^24 + 36*x^23 + 656*x^22 - 602*x^21 - 7532*x^20 + 5987*x^19 + 63242*x^18 - 39168*x^17 - 407978*x^16 + 177750*x^15 + 2044391*x^14 - 593805*x^13 - 7863783*x^12 + 1620832*x^11 + 22802242*x^10 - 3804296*x^9 - 48757198*x^8 + 7281003*x^7 + 74301119*x^6 - 12442481*x^5 - 74591007*x^4 + 16397553*x^3 + 41639217*x^2 - 9412671*x - 11716513, 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]])