# SageMath code for working with number field 38.0.2233638411813024816853081773648251688534529753590642239923912316757382599022775822751448518259.1
# 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^38 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023)
# 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^38 - 209*x^35 + 190*x^34 + 1292*x^33 + 11818*x^32 - 17860*x^31 - 124165*x^30 + 92378*x^29 - 315590*x^28 + 461434*x^27 + 19355661*x^26 + 42336256*x^25 - 170675214*x^24 - 2184021538*x^23 + 4055379323*x^22 + 20028771815*x^21 - 37281886089*x^20 - 119372774190*x^19 + 206222273796*x^18 + 638565435242*x^17 - 1510749792677*x^16 - 1246896758667*x^15 + 6615154002166*x^14 + 1497131351993*x^13 + 895955914113*x^12 - 6914578071407*x^11 + 51068057863185*x^10 + 80278712330599*x^9 - 10972845112675*x^8 + 78343381011053*x^7 + 257531197354487*x^6 + 248070995565179*x^5 + 172436840306532*x^4 + 97988777901672*x^3 + 65070800259551*x^2 + 24610290628033*x + 5454582062023)
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)]