# SageMath code for working with number field 45.15.1744174787462733669236743003323349838789896331385003649618101010724691.1

# 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 - 12*x^44 + 42*x^43 + 36*x^42 - 525*x^41 + 702*x^40 + 2050*x^39 - 5802*x^38 - 2832*x^37 + 26919*x^36 - 20811*x^35 - 75546*x^34 + 201699*x^33 - 97125*x^32 - 532302*x^31 + 1289591*x^30 - 477339*x^29 - 2405610*x^28 + 4128988*x^27 - 1666824*x^26 - 5259345*x^25 + 9273382*x^24 + 3749217*x^23 - 15223818*x^22 - 9661128*x^21 + 33692574*x^20 - 4282803*x^19 - 38096273*x^18 + 36480543*x^17 - 14154921*x^16 + 10925288*x^15 - 437979*x^14 - 36804246*x^13 + 57064132*x^12 - 25852146*x^11 - 34300866*x^10 + 48986404*x^9 - 2084457*x^8 - 23365995*x^7 + 5433983*x^6 + 5348430*x^5 - 1230861*x^4 - 665734*x^3 + 68376*x^2 + 38973*x + 2969)

# 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 - 12*x^44 + 42*x^43 + 36*x^42 - 525*x^41 + 702*x^40 + 2050*x^39 - 5802*x^38 - 2832*x^37 + 26919*x^36 - 20811*x^35 - 75546*x^34 + 201699*x^33 - 97125*x^32 - 532302*x^31 + 1289591*x^30 - 477339*x^29 - 2405610*x^28 + 4128988*x^27 - 1666824*x^26 - 5259345*x^25 + 9273382*x^24 + 3749217*x^23 - 15223818*x^22 - 9661128*x^21 + 33692574*x^20 - 4282803*x^19 - 38096273*x^18 + 36480543*x^17 - 14154921*x^16 + 10925288*x^15 - 437979*x^14 - 36804246*x^13 + 57064132*x^12 - 25852146*x^11 - 34300866*x^10 + 48986404*x^9 - 2084457*x^8 - 23365995*x^7 + 5433983*x^6 + 5348430*x^5 - 1230861*x^4 - 665734*x^3 + 68376*x^2 + 38973*x + 2969)
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)]