\\ Pari/GP code for working with number field 26.2.214593535987174144573161771120178694372847609970688.1

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

\\ Define the number field: 
K = bnfinit(y^26 - 13*y^25 + 121*y^24 - 802*y^23 + 4651*y^22 - 22825*y^21 + 98893*y^20 - 371962*y^19 + 1240872*y^18 - 3661594*y^17 + 9598744*y^16 - 22249906*y^15 + 46158003*y^14 - 86354433*y^13 + 145428273*y^12 - 217463318*y^11 + 270662252*y^10 - 252429562*y^9 + 127602004*y^8 + 63980066*y^7 - 189163921*y^6 + 167659735*y^5 - 112782103*y^4 + 89161818*y^3 - 55238589*y^2 + 18143595*y - 204972201, 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 - 13*x^25 + 121*x^24 - 802*x^23 + 4651*x^22 - 22825*x^21 + 98893*x^20 - 371962*x^19 + 1240872*x^18 - 3661594*x^17 + 9598744*x^16 - 22249906*x^15 + 46158003*x^14 - 86354433*x^13 + 145428273*x^12 - 217463318*x^11 + 270662252*x^10 - 252429562*x^9 + 127602004*x^8 + 63980066*x^7 - 189163921*x^6 + 167659735*x^5 - 112782103*x^4 + 89161818*x^3 - 55238589*x^2 + 18143595*x - 204972201, 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]])