# SageMath code for working with number field 44.0.860115008245742907292219227824365111518443501386754869093198564719785672888549376.3
# 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^44 + 3*x^42 + 9*x^40 + 27*x^38 + 81*x^36 + 243*x^34 + 729*x^32 + 2187*x^30 + 6561*x^28 + 19683*x^26 + 59049*x^24 + 177147*x^22 + 531441*x^20 + 1594323*x^18 + 4782969*x^16 + 14348907*x^14 + 43046721*x^12 + 129140163*x^10 + 387420489*x^8 + 1162261467*x^6 + 3486784401*x^4 + 10460353203*x^2 + 31381059609)
# 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^44 + 3*x^42 + 9*x^40 + 27*x^38 + 81*x^36 + 243*x^34 + 729*x^32 + 2187*x^30 + 6561*x^28 + 19683*x^26 + 59049*x^24 + 177147*x^22 + 531441*x^20 + 1594323*x^18 + 4782969*x^16 + 14348907*x^14 + 43046721*x^12 + 129140163*x^10 + 387420489*x^8 + 1162261467*x^6 + 3486784401*x^4 + 10460353203*x^2 + 31381059609)
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)]