# SageMath code for working with number field 18.6.51624177137754227336536682168566437749832086648167693412837925000000000000.1.
# 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^18 - 6*x^17 - 79635*x^16 + 726016*x^15 - 21346492116*x^14 + 262385243232*x^13 - 222640923218036*x^12 + 5287125117175824*x^11 + 36115874586042869742*x^10 - 315574620417688738476*x^9 + 1064246929788629129834742*x^8 - 10094756489315953842318336*x^7 + 5555830520763339794029338844*x^6 - 66983200453112231194458550608*x^5 - 95464018377851901223533667416996*x^4 + 244705662148773213221767445905616*x^3 - 887010722785755909642099142950505815*x^2 + 1957760134295180518228084208239496514*x - 371023633673372769266300083152445623019)

# 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^18 - 6*x^17 - 79635*x^16 + 726016*x^15 - 21346492116*x^14 + 262385243232*x^13 - 222640923218036*x^12 + 5287125117175824*x^11 + 36115874586042869742*x^10 - 315574620417688738476*x^9 + 1064246929788629129834742*x^8 - 10094756489315953842318336*x^7 + 5555830520763339794029338844*x^6 - 66983200453112231194458550608*x^5 - 95464018377851901223533667416996*x^4 + 244705662148773213221767445905616*x^3 - 887010722785755909642099142950505815*x^2 + 1957760134295180518228084208239496514*x - 371023633673372769266300083152445623019)
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 pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]