# SageMath code for working with number field 30.30.15010049358097810645468215169836895080663686428828497.1

# (Note that not all these functions may be available, and some may take a long time to execute.)

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^30 - x^29 - 52*x^28 + 51*x^27 + 1142*x^26 - 1092*x^25 - 13898*x^24 + 12856*x^23 + 103478*x^22 - 91664*x^21 - 491678*x^20 + 412467*x^19 + 1512708*x^18 - 1191563*x^17 - 3007832*x^16 + 2225799*x^15 + 3819257*x^14 - 2692049*x^13 - 3042474*x^12 + 2079355*x^11 + 1471733*x^10 - 992749*x^9 - 403229*x^8 + 274849*x^7 + 52970*x^6 - 38754*x^5 - 1790*x^4 + 2057*x^3 - 72*x^2 - 24*x + 1)

# Defining polynomial: 
K.defining_polynomial()

# Degree over Q: 
K.degree()

# Signature: 
K.signature()

# Discriminant: 
K.disc()

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

# 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()

# Galois group: 
K.galois_group(type='pari')

# Frobenius cycle types: 
p = 7;  # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
[(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]