# SageMath code for working with number field 42.0.124938828493629533000907736270182756286928467256210015044892719712262808512338377882811.1

# 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^41 + 231*x^40 - 1750*x^39 + 10395*x^38 - 52269*x^37 + 233611*x^36 - 952918*x^35 + 3599785*x^34 - 12710201*x^33 + 42287014*x^32 - 133370664*x^31 + 400618099*x^30 - 1149584975*x^29 + 3160451592*x^28 - 8341179427*x^27 + 21174580693*x^26 - 51755274760*x^25 + 121953514697*x^24 - 277155797093*x^23 + 608007380806*x^22 - 1287295716241*x^21 + 2631847521996*x^20 - 5192066875520*x^19 + 9886990587068*x^18 - 18146977046303*x^17 + 32112635555588*x^16 - 54655115299659*x^15 + 89504829240548*x^14 - 140479267770793*x^13 + 211514505374151*x^12 - 303526580264439*x^11 + 416119886096880*x^10 - 538840804150264*x^9 + 662937764461147*x^8 - 758558920220686*x^7 + 819087800747125*x^6 - 798550011094073*x^5 + 729734808064227*x^4 - 562691756062212*x^3 + 407486767661645*x^2 - 201294019183845*x + 99519182315771)

# 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^41 + 231*x^40 - 1750*x^39 + 10395*x^38 - 52269*x^37 + 233611*x^36 - 952918*x^35 + 3599785*x^34 - 12710201*x^33 + 42287014*x^32 - 133370664*x^31 + 400618099*x^30 - 1149584975*x^29 + 3160451592*x^28 - 8341179427*x^27 + 21174580693*x^26 - 51755274760*x^25 + 121953514697*x^24 - 277155797093*x^23 + 608007380806*x^22 - 1287295716241*x^21 + 2631847521996*x^20 - 5192066875520*x^19 + 9886990587068*x^18 - 18146977046303*x^17 + 32112635555588*x^16 - 54655115299659*x^15 + 89504829240548*x^14 - 140479267770793*x^13 + 211514505374151*x^12 - 303526580264439*x^11 + 416119886096880*x^10 - 538840804150264*x^9 + 662937764461147*x^8 - 758558920220686*x^7 + 819087800747125*x^6 - 798550011094073*x^5 + 729734808064227*x^4 - 562691756062212*x^3 + 407486767661645*x^2 - 201294019183845*x + 99519182315771)
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)]