# SageMath code for working with number field 45.45.18385118110130638532679220392530019242562666485797771487713623864599015675873914118339381361.1

# Some of these functions may take a long time to execute (this depends on the field).

# Define the number field: 
x = polygen(QQ);  K.<a> = NumberField(x^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*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()

# 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.<a> = NumberField(x^45 - 87*x^43 - 4*x^42 + 3402*x^41 + 300*x^40 - 79106*x^39 - 10008*x^38 + 1219668*x^37 + 196264*x^36 - 13165542*x^35 - 2519472*x^34 + 102436164*x^33 + 22316760*x^32 - 583045167*x^31 - 140145264*x^30 + 2440735434*x^29 + 631757436*x^28 - 7504722910*x^27 - 2050106328*x^26 + 16845457116*x^25 + 4771066968*x^24 - 27354225744*x^23 - 7899876240*x^22 + 31819110888*x^21 + 9218870784*x^20 - 26289431304*x^19 - 7517163552*x^18 + 15324352272*x^17 + 4245866400*x^16 - 6258116631*x^15 - 1643669037*x^14 + 1770836592*x^13 + 429144192*x^12 - 340508538*x^11 - 73586928*x^10 + 43022076*x^9 + 7934841*x^8 - 3375522*x^7 - 502012*x^6 + 149688*x^5 + 16758*x^4 - 3200*x^3 - 252*x^2 + 24*x + 1)
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)]