# SageMath code for working with number field 42.42.874571799455406731006354153891279294008499270793470105314249037985839659586368645179677.1

# 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 - 126*x^40 + 7371*x^38 - 265734*x^36 - 83*x^35 + 6608385*x^34 + 8715*x^33 - 120236886*x^32 - 418320*x^31 + 1656597096*x^30 + 12157425*x^29 - 17647825586*x^28 - 238834575*x^27 + 147151366404*x^26 + 3353237433*x^25 - 966133116900*x^24 - 34688663100*x^23 + 4999488494931*x^22 + 268659947305*x^21 - 20317109311296*x^20 - 1567171092690*x^19 + 64305659489211*x^18 + 6871145069880*x^17 - 156373597496214*x^16 - 22423210892208*x^15 + 286220369519701*x^14 + 53473028777910*x^13 - 382576512454206*x^12 - 90465142642875*x^11 + 356767740698292*x^10 + 103652112041415*x^9 - 215794576784304*x^8 - 74585905238228*x^7 + 74443140953685*x^6 + 29468446974645*x^5 - 11170399688028*x^4 - 4811855646528*x^3 + 319923920106*x^2 + 140495689782*x - 11523307067)

# 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 - 126*x^40 + 7371*x^38 - 265734*x^36 - 83*x^35 + 6608385*x^34 + 8715*x^33 - 120236886*x^32 - 418320*x^31 + 1656597096*x^30 + 12157425*x^29 - 17647825586*x^28 - 238834575*x^27 + 147151366404*x^26 + 3353237433*x^25 - 966133116900*x^24 - 34688663100*x^23 + 4999488494931*x^22 + 268659947305*x^21 - 20317109311296*x^20 - 1567171092690*x^19 + 64305659489211*x^18 + 6871145069880*x^17 - 156373597496214*x^16 - 22423210892208*x^15 + 286220369519701*x^14 + 53473028777910*x^13 - 382576512454206*x^12 - 90465142642875*x^11 + 356767740698292*x^10 + 103652112041415*x^9 - 215794576784304*x^8 - 74585905238228*x^7 + 74443140953685*x^6 + 29468446974645*x^5 - 11170399688028*x^4 - 4811855646528*x^3 + 319923920106*x^2 + 140495689782*x - 11523307067)
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)]