# SageMath code for working with number field 30.0.67540340288911750220987142641365428116063232133203.1

# 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^30 - x^29 + 12*x^28 - 45*x^27 + 210*x^26 - 744*x^25 + 2465*x^24 - 6965*x^23 + 19758*x^22 - 48686*x^21 + 121619*x^20 - 306546*x^19 + 810012*x^18 - 2087415*x^17 + 5002572*x^16 - 11311375*x^15 + 24424402*x^14 - 49257771*x^13 + 88022499*x^12 - 134078913*x^11 + 172501038*x^10 - 188691174*x^9 + 176841252*x^8 - 141293241*x^7 + 95362353*x^6 - 55362204*x^5 + 28414476*x^4 - 12173571*x^3 + 3774762*x^2 - 708588*x + 59049)

# 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^30 - x^29 + 12*x^28 - 45*x^27 + 210*x^26 - 744*x^25 + 2465*x^24 - 6965*x^23 + 19758*x^22 - 48686*x^21 + 121619*x^20 - 306546*x^19 + 810012*x^18 - 2087415*x^17 + 5002572*x^16 - 11311375*x^15 + 24424402*x^14 - 49257771*x^13 + 88022499*x^12 - 134078913*x^11 + 172501038*x^10 - 188691174*x^9 + 176841252*x^8 - 141293241*x^7 + 95362353*x^6 - 55362204*x^5 + 28414476*x^4 - 12173571*x^3 + 3774762*x^2 - 708588*x + 59049)
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)]