# SageMath code for working with number field 42.0.213117637842219136349783160187126852219651508028407629359584264706447783442638664863363.1

# 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 - 4*x^41 + 62*x^40 - 200*x^39 + 2132*x^38 - 6208*x^37 + 46584*x^36 - 115568*x^35 + 701352*x^34 - 1537594*x^33 + 7769821*x^32 - 14875116*x^31 + 64656220*x^30 - 109927490*x^29 + 414171141*x^28 - 618138014*x^27 + 2041345075*x^26 - 2683015728*x^25 + 7816607585*x^24 - 8902561137*x^23 + 22976431713*x^22 - 22760113866*x^21 + 52536783259*x^20 - 45434427911*x^19 + 92226880868*x^18 - 71336683352*x^17 + 125874997329*x^16 - 89801104142*x^15 + 131144820279*x^14 - 89289000509*x^13 + 106600416700*x^12 - 67935846970*x^11 + 65774842016*x^10 - 38731708226*x^9 + 30551737937*x^8 - 15767508635*x^7 + 9878705051*x^6 - 4297296085*x^5 + 2147704850*x^4 - 731044046*x^3 + 251922248*x^2 - 47762400*x + 8082649)

# 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 - 4*x^41 + 62*x^40 - 200*x^39 + 2132*x^38 - 6208*x^37 + 46584*x^36 - 115568*x^35 + 701352*x^34 - 1537594*x^33 + 7769821*x^32 - 14875116*x^31 + 64656220*x^30 - 109927490*x^29 + 414171141*x^28 - 618138014*x^27 + 2041345075*x^26 - 2683015728*x^25 + 7816607585*x^24 - 8902561137*x^23 + 22976431713*x^22 - 22760113866*x^21 + 52536783259*x^20 - 45434427911*x^19 + 92226880868*x^18 - 71336683352*x^17 + 125874997329*x^16 - 89801104142*x^15 + 131144820279*x^14 - 89289000509*x^13 + 106600416700*x^12 - 67935846970*x^11 + 65774842016*x^10 - 38731708226*x^9 + 30551737937*x^8 - 15767508635*x^7 + 9878705051*x^6 - 4297296085*x^5 + 2147704850*x^4 - 731044046*x^3 + 251922248*x^2 - 47762400*x + 8082649)
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)]