# SageMath code for working with number field 30.0.2666804038012396184155132908419351509268871501674291.1

# 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 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073)

# 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 - 5*x^29 + 23*x^28 - 62*x^27 + 198*x^26 - 367*x^25 + 397*x^24 + 1188*x^23 - 7770*x^22 + 27186*x^21 - 70571*x^20 + 106780*x^19 + 113195*x^18 - 1691169*x^17 + 8127778*x^16 - 29421234*x^15 + 90493414*x^14 - 242325966*x^13 + 570478488*x^12 - 1182396448*x^11 + 2168615679*x^10 - 3480425904*x^9 + 4808261059*x^8 - 5571332106*x^7 + 5293750647*x^6 - 4028704993*x^5 + 2404982206*x^4 - 1088744723*x^3 + 357808383*x^2 - 76833878*x + 8961073)
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)]