\\ Pari/GP code for working with number field 29.1.190430537333205962921320156798047381287357677729.1

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

\\ Define the number field: 
K = bnfinit(y^29 - 8*y^28 + 42*y^27 - 91*y^26 + 71*y^25 - 172*y^24 + 980*y^23 - 2477*y^22 + 4032*y^21 - 5998*y^20 + 8995*y^19 - 12396*y^18 + 15464*y^17 - 17477*y^16 + 17661*y^15 - 16777*y^14 + 14899*y^13 - 10749*y^12 + 6197*y^11 - 2594*y^10 - 1233*y^9 + 2318*y^8 - 113*y^7 - 855*y^6 + 1459*y^5 - 2625*y^4 + 2079*y^3 - 978*y^2 + 419*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^29 - 8*x^28 + 42*x^27 - 91*x^26 + 71*x^25 - 172*x^24 + 980*x^23 - 2477*x^22 + 4032*x^21 - 5998*x^20 + 8995*x^19 - 12396*x^18 + 15464*x^17 - 17477*x^16 + 17661*x^15 - 16777*x^14 + 14899*x^13 - 10749*x^12 + 6197*x^11 - 2594*x^10 - 1233*x^9 + 2318*x^8 - 113*x^7 - 855*x^6 + 1459*x^5 - 2625*x^4 + 2079*x^3 - 978*x^2 + 419*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]])