\\ Pari/GP code for working with number field 29.1.1310202274198394060556452494826429036980688154429108157810117051380399416309809.1

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

\\ Define the number field: 
K = bnfinit(y^29 - 13*y^28 - 55*y^27 + 712*y^26 + 11351*y^25 - 127986*y^24 + 66127*y^23 + 4301081*y^22 - 11864947*y^21 - 126534305*y^20 + 781815928*y^19 + 394279608*y^18 - 18262666572*y^17 + 39703600309*y^16 + 333330481344*y^15 - 1935249895004*y^14 + 1288520615067*y^13 + 24287341394015*y^12 - 64823993632866*y^11 - 202566749188801*y^10 + 1440509151901048*y^9 - 2780340418135933*y^8 - 4068418180429144*y^7 + 3309165651131592*y^6 + 104817181048914519*y^5 - 418797024905501732*y^4 + 871709514773509500*y^3 - 878388606425292930*y^2 + 5204661351059478078*y - 13446098969951310345, 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 - 13*x^28 - 55*x^27 + 712*x^26 + 11351*x^25 - 127986*x^24 + 66127*x^23 + 4301081*x^22 - 11864947*x^21 - 126534305*x^20 + 781815928*x^19 + 394279608*x^18 - 18262666572*x^17 + 39703600309*x^16 + 333330481344*x^15 - 1935249895004*x^14 + 1288520615067*x^13 + 24287341394015*x^12 - 64823993632866*x^11 - 202566749188801*x^10 + 1440509151901048*x^9 - 2780340418135933*x^8 - 4068418180429144*x^7 + 3309165651131592*x^6 + 104817181048914519*x^5 - 418797024905501732*x^4 + 871709514773509500*x^3 - 878388606425292930*x^2 + 5204661351059478078*x - 13446098969951310345, 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]])