# SageMath code for working with number field 45.45.99953767787456648146270917992226818804825612223555542932825373503718412411221640597660289227243073561.1

# 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 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919)

# 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 - 68*x^43 + 1390*x^42 - 278*x^41 - 68270*x^40 + 176199*x^39 + 1808389*x^38 - 8154574*x^37 - 26005805*x^36 + 196321268*x^35 + 127187382*x^34 - 2913805970*x^33 + 2272631340*x^32 + 27669670927*x^31 - 52510157887*x^30 - 162002621792*x^29 + 526806359978*x^28 + 468745083598*x^27 - 3175101889498*x^26 + 532391340063*x^25 + 12135569776697*x^24 - 10651958891479*x^23 - 28871893561060*x^22 + 45196495080913*x^21 + 38272655903792*x^20 - 105664143365844*x^19 - 13349385343870*x^18 + 153132315545987*x^17 - 41028745873427*x^16 - 141811047613612*x^15 + 76543731297893*x^14 + 83515595010770*x^13 - 65499266094722*x^12 - 29808531214812*x^11 + 32918818373161*x^10 + 5404646072895*x^9 - 10084991417837*x^8 - 16286468525*x^7 + 1821205338619*x^6 - 180698614131*x^5 - 173451418339*x^4 + 28592722199*x^3 + 6396948952*x^2 - 1303357845*x - 3078919)
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)]