\\ Pari/GP code for working with number field 24.4.91810054652229848026521481130092527303208363056182861328125.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^24 - 4*y^23 + 6*y^22 - 449*y^21 + 4273*y^20 + 5874*y^19 + 40584*y^18 - 1486656*y^17 + 320934*y^16 + 31758582*y^15 + 203615891*y^14 - 1041728624*y^13 - 8957494979*y^12 + 15522090746*y^11 + 187723100658*y^10 - 120260671411*y^9 - 2012872155141*y^8 + 555294801359*y^7 + 10666291776869*y^6 + 2324146207637*y^5 - 37689124873554*y^4 - 13595215964929*y^3 + 96703661945176*y^2 - 15091490947869*y - 91903199858837, 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

\\ Narrow class group: 
bnfnarrow(K)

\\ 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^24 - 4*x^23 + 6*x^22 - 449*x^21 + 4273*x^20 + 5874*x^19 + 40584*x^18 - 1486656*x^17 + 320934*x^16 + 31758582*x^15 + 203615891*x^14 - 1041728624*x^13 - 8957494979*x^12 + 15522090746*x^11 + 187723100658*x^10 - 120260671411*x^9 - 2012872155141*x^8 + 555294801359*x^7 + 10666291776869*x^6 + 2324146207637*x^5 - 37689124873554*x^4 - 13595215964929*x^3 + 96703661945176*x^2 - 15091490947869*x - 91903199858837, 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]])