\\ Pari/GP code for working with number field 37.37.81381208133441979421709122744091225498491936628940230588748580298513087650630871328595025812353503688138712627681.1

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

\\ Define the number field: 
K = bnfinit(y^37 - 666*y^35 - 481*y^34 + 193399*y^33 + 256521*y^32 - 32436827*y^31 - 60004084*y^30 + 3503452596*y^29 + 8113398850*y^28 - 257000630926*y^27 - 704292185114*y^26 + 13150853376643*y^25 + 41268101524916*y^24 - 474337598223255*y^23 - 1672482254287228*y^22 + 12032466243732809*y^21 + 47286637172470867*y^20 - 211502629431882231*y^19 - 929675982328753625*y^18 + 2498958523266151636*y^17 + 12536142236443615902*y^16 - 18791897342800992970*y^15 - 113239929214497210129*y^14 + 80927529107744817385*y^13 + 663966210336111429023*y^12 - 146546651897009713442*y^11 - 2439258160361411529393*y^10 - 154326326264692800324*y^9 + 5386184266991634344483*y^8 + 1055793633392402952768*y^7 - 6769150214799915444440*y^6 - 1552196159553019851407*y^5 + 4355671676506075568364*y^4 + 921754431147330625307*y^3 - 1108584774969076393499*y^2 - 217498966742521653325*y + 51140551819476687829, 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^37 - 666*x^35 - 481*x^34 + 193399*x^33 + 256521*x^32 - 32436827*x^31 - 60004084*x^30 + 3503452596*x^29 + 8113398850*x^28 - 257000630926*x^27 - 704292185114*x^26 + 13150853376643*x^25 + 41268101524916*x^24 - 474337598223255*x^23 - 1672482254287228*x^22 + 12032466243732809*x^21 + 47286637172470867*x^20 - 211502629431882231*x^19 - 929675982328753625*x^18 + 2498958523266151636*x^17 + 12536142236443615902*x^16 - 18791897342800992970*x^15 - 113239929214497210129*x^14 + 80927529107744817385*x^13 + 663966210336111429023*x^12 - 146546651897009713442*x^11 - 2439258160361411529393*x^10 - 154326326264692800324*x^9 + 5386184266991634344483*x^8 + 1055793633392402952768*x^7 - 6769150214799915444440*x^6 - 1552196159553019851407*x^5 + 4355671676506075568364*x^4 + 921754431147330625307*x^3 - 1108584774969076393499*x^2 - 217498966742521653325*x + 51140551819476687829, 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]])