# SageMath code for working with number field 36.36.22878331820822683097801634238807198405761132789439230168181892642289.1.
# 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^36 - 71*x^34 - 8*x^33 + 2203*x^32 + 458*x^31 - 39419*x^30 - 11171*x^29 + 452485*x^28 + 152549*x^27 - 3511489*x^26 - 1291483*x^25 + 18942377*x^24 + 7104729*x^23 - 72013209*x^22 - 25930789*x^21 + 193701628*x^20 + 63027785*x^19 - 366929496*x^18 - 100868241*x^17 + 482872954*x^16 + 103172083*x^15 - 430753452*x^14 - 63559658*x^13 + 250492399*x^12 + 20933394*x^11 - 89499925*x^10 - 2758730*x^9 + 18093496*x^8 - 12482*x^7 - 1913861*x^6 + 35416*x^5 + 94085*x^4 - 2330*x^3 - 1563*x^2 + 15*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^36 - 71*x^34 - 8*x^33 + 2203*x^32 + 458*x^31 - 39419*x^30 - 11171*x^29 + 452485*x^28 + 152549*x^27 - 3511489*x^26 - 1291483*x^25 + 18942377*x^24 + 7104729*x^23 - 72013209*x^22 - 25930789*x^21 + 193701628*x^20 + 63027785*x^19 - 366929496*x^18 - 100868241*x^17 + 482872954*x^16 + 103172083*x^15 - 430753452*x^14 - 63559658*x^13 + 250492399*x^12 + 20933394*x^11 - 89499925*x^10 - 2758730*x^9 + 18093496*x^8 - 12482*x^7 - 1913861*x^6 + 35416*x^5 + 94085*x^4 - 2330*x^3 - 1563*x^2 + 15*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 pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]