\\ Pari/GP code for working with number field 29.29.158165571476523684791187605131317359442300677347381843010071500056545201.1

\\ 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 - 168*y^27 + 353*y^26 + 11480*y^25 - 34328*y^24 - 410190*y^23 + 1556443*y^22 + 8267052*y^21 - 38827232*y^20 - 94788585*y^19 + 575669857*y^18 + 574148959*y^17 - 5286405734*y^16 - 1118675686*y^15 + 30698615326*y^14 - 6935797323*y^13 - 112817035992*y^12 + 51055206761*y^11 + 256367413593*y^10 - 136472035194*y^9 - 340428884896*y^8 + 170672875682*y^7 + 236550118296*y^6 - 94550319984*y^5 - 69592403677*y^4 + 18732474566*y^3 + 3480965541*y^2 - 875365500*y + 10242329, 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 - 168*x^27 + 353*x^26 + 11480*x^25 - 34328*x^24 - 410190*x^23 + 1556443*x^22 + 8267052*x^21 - 38827232*x^20 - 94788585*x^19 + 575669857*x^18 + 574148959*x^17 - 5286405734*x^16 - 1118675686*x^15 + 30698615326*x^14 - 6935797323*x^13 - 112817035992*x^12 + 51055206761*x^11 + 256367413593*x^10 - 136472035194*x^9 - 340428884896*x^8 + 170672875682*x^7 + 236550118296*x^6 - 94550319984*x^5 - 69592403677*x^4 + 18732474566*x^3 + 3480965541*x^2 - 875365500*x + 10242329, 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]])