\\ Pari/GP code for working with number field 42.0.2281836760183646137444154412268560109828024514076489472840222217265158917203.1

\\ (Note that not all these functions may be available, and some may take a long time to execute.)

\\ Define the number field: 
K = bnfinit(x^42 - x^41 + 21*x^40 - 18*x^39 + 248*x^38 - 191*x^37 + 1996*x^36 - 1375*x^35 + 12088*x^34 - 7495*x^33 + 57373*x^32 - 31825*x^31 + 219409*x^30 - 108700*x^29 + 684645*x^28 - 300153*x^27 + 1757705*x^26 - 677756*x^25 + 3715466*x^24 - 1242500*x^23 + 6455978*x^22 - 1853597*x^21 + 9148985*x^20 - 2205194*x^19 + 10469894*x^18 - 2090336*x^17 + 9505112*x^16 - 1507895*x^15 + 6709547*x^14 - 839645*x^13 + 3559478*x^12 - 314951*x^11 + 1368367*x^10 - 93808*x^9 + 355278*x^8 - 10362*x^7 + 58036*x^6 - 2497*x^5 + 4950*x^4 + 165*x^3 + 176*x^2 - 11*x + 1, 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

\\ Galois group: 
polgalois(K.pol)

\\ Frobenius cycle types: 
p = 7;  \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
idealfactors = idealprimedec(K, p); \\ get the data
vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])