\\ Pari/GP code for working with number field 26.26.294598768324608440893618976324163386033790283203125.1

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

\\ Define the number field: 
K = bnfinit(y^26 - 11*y^25 - 7*y^24 + 468*y^23 - 1046*y^22 - 6960*y^21 + 26796*y^20 + 40357*y^19 - 278503*y^18 + 355*y^17 + 1499785*y^16 - 1077692*y^15 - 4479644*y^14 + 5284410*y^13 + 7470556*y^12 - 11963101*y^11 - 6678940*y^10 + 14762504*y^9 + 2744933*y^8 - 10366844*y^7 - 29320*y^6 + 4079010*y^5 - 378384*y^4 - 821546*y^3 + 120702*y^2 + 64121*y - 11789, 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 - 11*x^25 - 7*x^24 + 468*x^23 - 1046*x^22 - 6960*x^21 + 26796*x^20 + 40357*x^19 - 278503*x^18 + 355*x^17 + 1499785*x^16 - 1077692*x^15 - 4479644*x^14 + 5284410*x^13 + 7470556*x^12 - 11963101*x^11 - 6678940*x^10 + 14762504*x^9 + 2744933*x^8 - 10366844*x^7 - 29320*x^6 + 4079010*x^5 - 378384*x^4 - 821546*x^3 + 120702*x^2 + 64121*x - 11789, 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]])