# SageMath code for working with number field 18.18.389573518709461277048874506777900581813141954677569649354154692963827395433931232826260909975966492327717495503836697626432438146795529306112.1.
# 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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312)

# 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 - 146687112*x^16 - 37892719296*x^15 + 8408618911888512*x^14 + 4325813678364681744*x^13 - 243604861180754026285992*x^12 - 192688273988213296086844512*x^11 + 3797668582351192771935287035158*x^10 + 4155203541072329873982601967541328*x^9 - 30824703601196378502940767667441818360*x^8 - 44117476415146499319729949379490251620992*x^7 + 112503359565037178406650855732715578314837728*x^6 + 209476226244754241024858216530145737787244951984*x^5 - 121787795931109896131166351794082151507555501395576*x^4 - 371916116695980755216640900185153166330681546329851680*x^3 - 57935826570177631099916455044945775522459483676836854543*x^2 + 201710684116538249174185560612387919784188057130327410730544*x + 93572448667401892834025834694441537328078001525771364657373312)
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)]