# SageMath code for working with number field 27.27.1670605670104664083337543234069150946215895123101528358912.1.
# 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^27 - 78*x^25 - 69*x^24 + 2391*x^23 + 3729*x^22 - 36776*x^21 - 78168*x^20 + 303360*x^19 + 833056*x^18 - 1289328*x^17 - 4920498*x^16 + 1998961*x^15 + 16364808*x^14 + 3861165*x^13 - 29580077*x^12 - 18957114*x^11 + 26856084*x^10 + 25541057*x^9 - 11892138*x^8 - 16024680*x^7 + 2411455*x^6 + 5218554*x^5 - 217893*x^4 - 866850*x^3 + 37389*x^2 + 59427*x - 7561)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

# Ramified primes: 
K.disc().support()

# Automorphisms: 
K.automorphisms()

# Integral basis: 
K.integral_basis()

# Class group: 
K.class_group().invariants()

# Narrow class group: 
K.narrow_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^27 - 78*x^25 - 69*x^24 + 2391*x^23 + 3729*x^22 - 36776*x^21 - 78168*x^20 + 303360*x^19 + 833056*x^18 - 1289328*x^17 - 4920498*x^16 + 1998961*x^15 + 16364808*x^14 + 3861165*x^13 - 29580077*x^12 - 18957114*x^11 + 26856084*x^10 + 25541057*x^9 - 11892138*x^8 - 16024680*x^7 + 2411455*x^6 + 5218554*x^5 - 217893*x^4 - 866850*x^3 + 37389*x^2 + 59427*x - 7561)
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()

# 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)]