# SageMath code for working with number field 42.42.104017955712751803355033526522081856753017553948018605377854818344593048095703125.1

# 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 - 61*x^40 + 56*x^39 + 1654*x^38 - 1379*x^37 - 26380*x^36 + 19735*x^35 + 276302*x^34 - 183047*x^33 - 2012061*x^32 + 1164081*x^31 + 10536175*x^30 - 5248100*x^29 - 40536685*x^28 + 17139885*x^27 + 116229751*x^26 - 41141876*x^25 - 250648402*x^24 + 73243022*x^23 + 408521488*x^22 - 97102633*x^21 - 503598537*x^20 + 95768202*x^19 + 467783098*x^18 - 69790048*x^17 - 324502952*x^16 + 37071837*x^15 + 165565463*x^14 - 14037193*x^13 - 60679502*x^12 + 3661307*x^11 + 15419833*x^10 - 625768*x^9 - 2575782*x^8 + 66132*x^7 + 260062*x^6 - 4477*x^5 - 13750*x^4 + 275*x^3 + 286*x^2 - 11*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 - x^41 - 61*x^40 + 56*x^39 + 1654*x^38 - 1379*x^37 - 26380*x^36 + 19735*x^35 + 276302*x^34 - 183047*x^33 - 2012061*x^32 + 1164081*x^31 + 10536175*x^30 - 5248100*x^29 - 40536685*x^28 + 17139885*x^27 + 116229751*x^26 - 41141876*x^25 - 250648402*x^24 + 73243022*x^23 + 408521488*x^22 - 97102633*x^21 - 503598537*x^20 + 95768202*x^19 + 467783098*x^18 - 69790048*x^17 - 324502952*x^16 + 37071837*x^15 + 165565463*x^14 - 14037193*x^13 - 60679502*x^12 + 3661307*x^11 + 15419833*x^10 - 625768*x^9 - 2575782*x^8 + 66132*x^7 + 260062*x^6 - 4477*x^5 - 13750*x^4 + 275*x^3 + 286*x^2 - 11*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)]