# SageMath code for working with number field 30.0.8851375005191383462430321349782942769050796640392327.1

# 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 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689)

# 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 - 4*x^29 - 32*x^28 + 92*x^27 + 588*x^26 - 752*x^25 - 6717*x^24 - 805*x^23 + 44049*x^22 + 54263*x^21 - 93757*x^20 - 350454*x^19 - 591742*x^18 + 307201*x^17 + 3600412*x^16 + 5669890*x^15 - 632841*x^14 - 17605194*x^13 - 34361042*x^12 - 23087625*x^11 + 45815379*x^10 + 119941341*x^9 + 85868062*x^8 - 30516460*x^7 - 62011703*x^6 + 46150369*x^5 + 155790202*x^4 + 147686157*x^3 + 67768188*x^2 + 10788948*x + 751689)
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)]