# SageMath code for working with number field 42.42.496897759422042196258605771077406782550407598249513303021389442457964675897236469.1

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

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^42 - x^41 - 67*x^40 + 62*x^39 + 1993*x^38 - 1704*x^37 - 34847*x^36 + 27519*x^35 + 399768*x^34 - 291838*x^33 - 3181854*x^32 + 2151119*x^31 + 18125561*x^30 - 11376768*x^29 - 75199314*x^28 + 43948028*x^27 + 229267037*x^26 - 125109956*x^25 - 515411960*x^24 + 263200585*x^23 + 854114535*x^22 - 408541162*x^21 - 1040280967*x^20 + 465602383*x^19 + 926307827*x^18 - 386475878*x^17 - 597992464*x^16 + 230741151*x^15 + 276295754*x^14 - 97190872*x^13 - 89602671*x^12 + 28039534*x^11 + 19807180*x^10 - 5297092*x^9 - 2853616*x^8 + 611802*x^7 + 248962*x^6 - 38836*x^5 - 11508*x^4 + 1155*x^3 + 217*x^2 - 14*x - 1)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

# Ramified primes: 
K.disc().support()

# Autmorphisms: 
K.automorphisms()

# Integral basis: 
K.integral_basis()

# Class group: 
K.class_group().invariants()

# Unit group: 
UK = K.unit_group()

# Unit rank: 
UK.rank()

# Generator for roots of unity: 
UK.torsion_generator()

# Fundamental units: 
UK.fundamental_units()

# Regulator: 
K.regulator()

# Analytic class number formula: 
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ);  K.<a> = NumberField(x^42 - x^41 - 67*x^40 + 62*x^39 + 1993*x^38 - 1704*x^37 - 34847*x^36 + 27519*x^35 + 399768*x^34 - 291838*x^33 - 3181854*x^32 + 2151119*x^31 + 18125561*x^30 - 11376768*x^29 - 75199314*x^28 + 43948028*x^27 + 229267037*x^26 - 125109956*x^25 - 515411960*x^24 + 263200585*x^23 + 854114535*x^22 - 408541162*x^21 - 1040280967*x^20 + 465602383*x^19 + 926307827*x^18 - 386475878*x^17 - 597992464*x^16 + 230741151*x^15 + 276295754*x^14 - 97190872*x^13 - 89602671*x^12 + 28039534*x^11 + 19807180*x^10 - 5297092*x^9 - 2853616*x^8 + 611802*x^7 + 248962*x^6 - 38836*x^5 - 11508*x^4 + 1155*x^3 + 217*x^2 - 14*x - 1)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator();  RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))  

# Intermediate fields: 
K.subfields()[1:-1]

# Galois group: 
K.galois_group(type='pari')

# 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 Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]