\\ Pari/GP code for working with number field 21.21.988509426193021817935320621873627136.1

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

\\ Define the number field: 
K = bnfinit(y^21 - 7*y^20 - 45*y^19 + 362*y^18 + 799*y^17 - 7508*y^16 - 7656*y^15 + 79875*y^14 + 49998*y^13 - 459906*y^12 - 252543*y^11 + 1367403*y^10 + 782753*y^9 - 1809622*y^8 - 897807*y^7 + 1116746*y^6 + 430780*y^5 - 303555*y^4 - 87913*y^3 + 25235*y^2 + 7546*y + 343, 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^21 - 7*x^20 - 45*x^19 + 362*x^18 + 799*x^17 - 7508*x^16 - 7656*x^15 + 79875*x^14 + 49998*x^13 - 459906*x^12 - 252543*x^11 + 1367403*x^10 + 782753*x^9 - 1809622*x^8 - 897807*x^7 + 1116746*x^6 + 430780*x^5 - 303555*x^4 - 87913*x^3 + 25235*x^2 + 7546*x + 343, 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]])