\\ Pari/GP code for working with number field 28.28.2311051677985488109296342199847259892756218483742420538210789.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^28 - y^27 - 231*y^26 + 231*y^25 + 23897*y^24 - 23897*y^23 - 1460903*y^22 + 1460903*y^21 + 58643801*y^20 - 58643801*y^19 - 1624287911*y^18 + 1624287911*y^17 + 31741662553*y^16 - 31741662553*y^15 - 439714884263*y^14 + 439714884263*y^13 + 4274850583897*y^12 - 4274850583897*y^11 - 28412803328679*y^10 + 28412803328679*y^9 + 122982646371673*y^8 - 122982646371673*y^7 - 317440480029351*y^6 + 317440480029351*y^5 + 407962316395865*y^4 - 407962316395865*y^3 - 150039834700455*y^2 + 150039834700455*y - 22496485878439, 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^28 - x^27 - 231*x^26 + 231*x^25 + 23897*x^24 - 23897*x^23 - 1460903*x^22 + 1460903*x^21 + 58643801*x^20 - 58643801*x^19 - 1624287911*x^18 + 1624287911*x^17 + 31741662553*x^16 - 31741662553*x^15 - 439714884263*x^14 + 439714884263*x^13 + 4274850583897*x^12 - 4274850583897*x^11 - 28412803328679*x^10 + 28412803328679*x^9 + 122982646371673*x^8 - 122982646371673*x^7 - 317440480029351*x^6 + 317440480029351*x^5 + 407962316395865*x^4 - 407962316395865*x^3 - 150039834700455*x^2 + 150039834700455*x - 22496485878439, 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(L)]

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])