# SageMath code for working with number field 42.0.16778739246697564329550246936340186720321059686137149293129858772813957238199098503.1

# 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 + 13*x^40 - 18*x^39 + 139*x^38 - 250*x^37 + 1451*x^36 + 321*x^35 + 11819*x^34 + 7069*x^33 + 101147*x^32 + 34723*x^31 + 906407*x^30 - 143662*x^29 + 2718926*x^28 - 434521*x^27 + 7093735*x^26 - 1731307*x^25 + 17818760*x^24 - 17036040*x^23 + 50814721*x^22 - 44435389*x^21 + 124550915*x^20 - 97201554*x^19 + 258557330*x^18 - 191477895*x^17 + 222793332*x^16 - 183429282*x^15 + 205234539*x^14 - 104619060*x^13 + 150123312*x^12 + 66144604*x^11 + 21062459*x^10 + 5929237*x^9 + 1586736*x^8 + 394344*x^7 + 106707*x^6 + 17635*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*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 + 13*x^40 - 18*x^39 + 139*x^38 - 250*x^37 + 1451*x^36 + 321*x^35 + 11819*x^34 + 7069*x^33 + 101147*x^32 + 34723*x^31 + 906407*x^30 - 143662*x^29 + 2718926*x^28 - 434521*x^27 + 7093735*x^26 - 1731307*x^25 + 17818760*x^24 - 17036040*x^23 + 50814721*x^22 - 44435389*x^21 + 124550915*x^20 - 97201554*x^19 + 258557330*x^18 - 191477895*x^17 + 222793332*x^16 - 183429282*x^15 + 205234539*x^14 - 104619060*x^13 + 150123312*x^12 + 66144604*x^11 + 21062459*x^10 + 5929237*x^9 + 1586736*x^8 + 394344*x^7 + 106707*x^6 + 17635*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*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)]