# SageMath code for working with number field 45.45.16527441490674381067348948889146522048718287580785548225593089356497792068001085725808918264262962961.1
# 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^45 - 4*x^44 - 114*x^43 + 468*x^42 + 5696*x^41 - 24132*x^40 - 165566*x^39 + 729869*x^38 + 3130045*x^37 - 14526130*x^36 - 40702810*x^35 + 202181480*x^34 + 374739344*x^33 - 2040701711*x^32 - 2468141099*x^31 + 15280860430*x^30 + 11523689449*x^29 - 86102308596*x^28 - 36476021474*x^27 + 368042106614*x^26 + 66266177672*x^25 - 1197228116258*x^24 + 511532340*x^23 + 2959992694385*x^22 - 394556499523*x^21 - 5529580093472*x^20 + 1265687493078*x^19 + 7719490702430*x^18 - 2271295840968*x^17 - 7914078560767*x^16 + 2645087732416*x^15 + 5804631677810*x^14 - 2048306537737*x^13 - 2929193071000*x^12 + 1039444940408*x^11 + 957147248584*x^10 - 333732759991*x^9 - 182597145321*x^8 + 64595300484*x^7 + 16257686900*x^6 - 6916885901*x^5 - 166352258*x^4 + 298954277*x^3 - 40977465*x^2 + 2118031*x - 36851)
# 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^45 - 4*x^44 - 114*x^43 + 468*x^42 + 5696*x^41 - 24132*x^40 - 165566*x^39 + 729869*x^38 + 3130045*x^37 - 14526130*x^36 - 40702810*x^35 + 202181480*x^34 + 374739344*x^33 - 2040701711*x^32 - 2468141099*x^31 + 15280860430*x^30 + 11523689449*x^29 - 86102308596*x^28 - 36476021474*x^27 + 368042106614*x^26 + 66266177672*x^25 - 1197228116258*x^24 + 511532340*x^23 + 2959992694385*x^22 - 394556499523*x^21 - 5529580093472*x^20 + 1265687493078*x^19 + 7719490702430*x^18 - 2271295840968*x^17 - 7914078560767*x^16 + 2645087732416*x^15 + 5804631677810*x^14 - 2048306537737*x^13 - 2929193071000*x^12 + 1039444940408*x^11 + 957147248584*x^10 - 333732759991*x^9 - 182597145321*x^8 + 64595300484*x^7 + 16257686900*x^6 - 6916885901*x^5 - 166352258*x^4 + 298954277*x^3 - 40977465*x^2 + 2118031*x - 36851)
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)]