\\ Pari/GP code for working with number field 45.45.690502119755999041650933728181031670984689765738417760130764409811234288886886833270367975576584121365281.1.
\\ Some of these functions may take a long time to execute (this depends on the field).


\\ Define the number field: 
K = bnfinit(y^45 - 4*y^44 - 146*y^43 + 698*y^42 + 8985*y^41 - 51888*y^40 - 296427*y^39 + 2177976*y^38 + 5305247*y^37 - 57476954*y^36 - 34012820*y^35 + 1002110429*y^34 - 626632964*y^33 - 11753633659*y^32 + 18688879395*y^31 + 91599636078*y^30 - 237738671647*y^29 - 440115277508*y^28 + 1861290436646*y^27 + 876346054868*y^26 - 9636797401413*y^25 + 3638557275086*y^24 + 33031864898329*y^23 - 35155604700368*y^22 - 70196515943930*y^21 + 133850962002890*y^20 + 68063148815178*y^19 - 296060971462243*y^18 + 63115875057640*y^17 + 386991855136149*y^16 - 292070548579474*y^15 - 252378957195025*y^14 + 393110942284862*y^13 - 3989425272418*y^12 - 248373295675609*y^11 + 116468486134606*y^10 + 55635812822046*y^9 - 63558461838326*y^8 + 10039317831256*y^7 + 9501878099079*y^6 - 4834005607020*y^5 + 527330092663*y^4 + 166660835109*y^3 - 51025747438*y^2 + 4407259697*y - 98524081, 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^45 - 4*x^44 - 146*x^43 + 698*x^42 + 8985*x^41 - 51888*x^40 - 296427*x^39 + 2177976*x^38 + 5305247*x^37 - 57476954*x^36 - 34012820*x^35 + 1002110429*x^34 - 626632964*x^33 - 11753633659*x^32 + 18688879395*x^31 + 91599636078*x^30 - 237738671647*x^29 - 440115277508*x^28 + 1861290436646*x^27 + 876346054868*x^26 - 9636797401413*x^25 + 3638557275086*x^24 + 33031864898329*x^23 - 35155604700368*x^22 - 70196515943930*x^21 + 133850962002890*x^20 + 68063148815178*x^19 - 296060971462243*x^18 + 63115875057640*x^17 + 386991855136149*x^16 - 292070548579474*x^15 - 252378957195025*x^14 + 393110942284862*x^13 - 3989425272418*x^12 - 248373295675609*x^11 + 116468486134606*x^10 + 55635812822046*x^9 - 63558461838326*x^8 + 10039317831256*x^7 + 9501878099079*x^6 - 4834005607020*x^5 + 527330092663*x^4 + 166660835109*x^3 - 51025747438*x^2 + 4407259697*x - 98524081, 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]])