\\ Pari/GP code for working with number field 29.29.38358032782038398419973086399760468678777161743121.1

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

\\ Define the number field: 
K = bnfinit(y^29 - y^28 - 28*y^27 + 27*y^26 + 351*y^25 - 325*y^24 - 2600*y^23 + 2300*y^22 + 12650*y^21 - 10626*y^20 - 42504*y^19 + 33649*y^18 + 100947*y^17 - 74613*y^16 - 170544*y^15 + 116280*y^14 + 203490*y^13 - 125970*y^12 - 167960*y^11 + 92378*y^10 + 92378*y^9 - 43758*y^8 - 31824*y^7 + 12376*y^6 + 6188*y^5 - 1820*y^4 - 560*y^3 + 105*y^2 + 15*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 - x^28 - 28*x^27 + 27*x^26 + 351*x^25 - 325*x^24 - 2600*x^23 + 2300*x^22 + 12650*x^21 - 10626*x^20 - 42504*x^19 + 33649*x^18 + 100947*x^17 - 74613*x^16 - 170544*x^15 + 116280*x^14 + 203490*x^13 - 125970*x^12 - 167960*x^11 + 92378*x^10 + 92378*x^9 - 43758*x^8 - 31824*x^7 + 12376*x^6 + 6188*x^5 - 1820*x^4 - 560*x^3 + 105*x^2 + 15*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]])