# SageMath code for working with number field 42.0.176602720807616761537805583365440112858555316650025456145851095351761290003.1

# 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 + 21*x^40 + 252*x^38 + 2065*x^36 - x^35 + 12789*x^34 - 7*x^33 + 62181*x^32 + 7*x^31 + 244265*x^30 + 574*x^29 + 784101*x^28 + 5341*x^27 + 2075577*x^26 + 29988*x^25 + 4529364*x^24 + 118188*x^23 + 8142834*x^22 + 346087*x^21 + 11951541*x^20 + 767956*x^19 + 14210490*x^18 + 1288070*x^17 + 13430361*x^16 + 1624252*x^15 + 9960725*x^14 + 1486219*x^13 + 5587183*x^12 + 983626*x^11 + 2322670*x^10 + 429093*x^9 + 648396*x^8 + 121077*x^7 + 117649*x^6 + 14763*x^5 + 7399*x^4 + 56*x^3 + 245*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 + 21*x^40 + 252*x^38 + 2065*x^36 - x^35 + 12789*x^34 - 7*x^33 + 62181*x^32 + 7*x^31 + 244265*x^30 + 574*x^29 + 784101*x^28 + 5341*x^27 + 2075577*x^26 + 29988*x^25 + 4529364*x^24 + 118188*x^23 + 8142834*x^22 + 346087*x^21 + 11951541*x^20 + 767956*x^19 + 14210490*x^18 + 1288070*x^17 + 13430361*x^16 + 1624252*x^15 + 9960725*x^14 + 1486219*x^13 + 5587183*x^12 + 983626*x^11 + 2322670*x^10 + 429093*x^9 + 648396*x^8 + 121077*x^7 + 117649*x^6 + 14763*x^5 + 7399*x^4 + 56*x^3 + 245*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)]