# SageMath code for working with number field 42.0.4472772095648327544266441640449519840379754819764800031247757188817501068115234375.1

# 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 + 44*x^40 - 44*x^39 + 904*x^38 - 904*x^37 + 11525*x^36 - 11525*x^35 + 102212*x^34 - 102212*x^33 + 670199*x^32 - 670199*x^31 + 3371975*x^30 - 3371975*x^29 + 13342815*x^28 - 13342815*x^27 + 42258251*x^26 - 42258251*x^25 + 108593663*x^24 - 108593663*x^23 + 229203503*x^22 - 229203503*x^21 + 402580148*x^20 - 402580148*x^19 + 598327973*x^18 - 598327973*x^17 + 769983758*x^16 - 769983758*x^15 + 884984678*x^14 - 884984678*x^13 + 942485138*x^12 - 942485138*x^11 + 963249193*x^10 - 963249193*x^9 + 968416718*x^8 - 968416718*x^7 + 969243522*x^6 - 969243522*x^5 + 969319675*x^4 - 969319675*x^3 + 969322986*x^2 - 969322986*x + 969323029)

# 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 + 44*x^40 - 44*x^39 + 904*x^38 - 904*x^37 + 11525*x^36 - 11525*x^35 + 102212*x^34 - 102212*x^33 + 670199*x^32 - 670199*x^31 + 3371975*x^30 - 3371975*x^29 + 13342815*x^28 - 13342815*x^27 + 42258251*x^26 - 42258251*x^25 + 108593663*x^24 - 108593663*x^23 + 229203503*x^22 - 229203503*x^21 + 402580148*x^20 - 402580148*x^19 + 598327973*x^18 - 598327973*x^17 + 769983758*x^16 - 769983758*x^15 + 884984678*x^14 - 884984678*x^13 + 942485138*x^12 - 942485138*x^11 + 963249193*x^10 - 963249193*x^9 + 968416718*x^8 - 968416718*x^7 + 969243522*x^6 - 969243522*x^5 + 969319675*x^4 - 969319675*x^3 + 969322986*x^2 - 969322986*x + 969323029)
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)]