# SageMath code for working with number field 30.2.114463058803512608608063235872919561806671142578125.1
# Some of these functions may take a long time to execute (this depends on the field).
# Define the number field:
x = polygen(QQ); K. = NumberField(x^30 - 3*x^29 + 9*x^28 + 14*x^27 - 113*x^26 + 370*x^25 - 643*x^24 - 15*x^23 + 3880*x^22 - 13302*x^21 + 28770*x^20 - 44390*x^19 + 53641*x^18 - 46319*x^17 + 35868*x^16 - 26390*x^15 + 12095*x^14 + 43878*x^13 + 45313*x^12 - 6914*x^11 - 59831*x^10 - 94496*x^9 - 64150*x^8 - 2932*x^7 + 54993*x^6 + 69070*x^5 + 52989*x^4 + 26576*x^3 + 8569*x^2 + 484*x - 121)
# 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. = NumberField(x^30 - 3*x^29 + 9*x^28 + 14*x^27 - 113*x^26 + 370*x^25 - 643*x^24 - 15*x^23 + 3880*x^22 - 13302*x^21 + 28770*x^20 - 44390*x^19 + 53641*x^18 - 46319*x^17 + 35868*x^16 - 26390*x^15 + 12095*x^14 + 43878*x^13 + 45313*x^12 - 6914*x^11 - 59831*x^10 - 94496*x^9 - 64150*x^8 - 2932*x^7 + 54993*x^6 + 69070*x^5 + 52989*x^4 + 26576*x^3 + 8569*x^2 + 484*x - 121)
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)]