# SageMath code for working with number field 42.0.121842012423466724043945276342536999203112328184628981033554149680639161638328200007.1

# 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 + 62*x^40 - 55*x^39 + 1885*x^38 - 1493*x^37 + 36890*x^36 - 26096*x^35 + 516884*x^34 - 325868*x^33 + 5474760*x^32 - 3063183*x^31 + 45214327*x^30 - 22319399*x^29 + 296463014*x^28 - 128129100*x^27 + 1558565095*x^26 - 584173775*x^25 + 6595257074*x^24 - 2118905803*x^23 + 22443263173*x^22 - 6093288301*x^21 + 61089146298*x^20 - 13772999112*x^19 + 131703790552*x^18 - 24134677936*x^17 + 221581028128*x^16 - 32118789120*x^15 + 284876509184*x^14 - 31567940608*x^13 + 271846162432*x^12 - 21964134400*x^11 + 184905250816*x^10 - 10296512512*x^9 + 84546772992*x^8 - 2848260096*x^7 + 23814799360*x^6 - 531300352*x^5 + 3554017280*x^4 + 28835840*x^3 + 213385216*x^2 - 11534336*x + 2097152)

# 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 + 62*x^40 - 55*x^39 + 1885*x^38 - 1493*x^37 + 36890*x^36 - 26096*x^35 + 516884*x^34 - 325868*x^33 + 5474760*x^32 - 3063183*x^31 + 45214327*x^30 - 22319399*x^29 + 296463014*x^28 - 128129100*x^27 + 1558565095*x^26 - 584173775*x^25 + 6595257074*x^24 - 2118905803*x^23 + 22443263173*x^22 - 6093288301*x^21 + 61089146298*x^20 - 13772999112*x^19 + 131703790552*x^18 - 24134677936*x^17 + 221581028128*x^16 - 32118789120*x^15 + 284876509184*x^14 - 31567940608*x^13 + 271846162432*x^12 - 21964134400*x^11 + 184905250816*x^10 - 10296512512*x^9 + 84546772992*x^8 - 2848260096*x^7 + 23814799360*x^6 - 531300352*x^5 + 3554017280*x^4 + 28835840*x^3 + 213385216*x^2 - 11534336*x + 2097152)
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)]