# SageMath code for working with number field 45.45.299215681303998835585125432825671967739342947202402933846152911778748517690473818220198154449462890625.1

# 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 - 5*x^44 - 120*x^43 + 570*x^42 + 6560*x^41 - 29130*x^40 - 216795*x^39 + 884935*x^38 + 4846835*x^37 - 17873115*x^36 - 77748559*x^35 + 254358990*x^34 + 926664900*x^33 - 2637505030*x^32 - 8389061335*x^31 + 20318079721*x^30 + 58494148730*x^29 - 117428976980*x^28 - 316667682495*x^27 + 510052425895*x^26 + 1335050206763*x^25 - 1653394093375*x^24 - 4375338382740*x^23 + 3923851568060*x^22 + 11069977015865*x^21 - 6532787291841*x^20 - 21348397264600*x^19 + 6845293868685*x^18 + 30760204370920*x^17 - 2712180162090*x^16 - 32145143355065*x^15 - 3532224571660*x^14 + 23314933370325*x^13 + 6500465979650*x^12 - 10976198292280*x^11 - 4703287953351*x^10 + 3011509345140*x^9 + 1752830160915*x^8 - 400149323895*x^7 - 327698710405*x^6 + 19849150584*x^5 + 30510229725*x^4 + 7497705*x^3 - 1362119415*x^2 - 14453695*x + 22808701)

# 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 - 5*x^44 - 120*x^43 + 570*x^42 + 6560*x^41 - 29130*x^40 - 216795*x^39 + 884935*x^38 + 4846835*x^37 - 17873115*x^36 - 77748559*x^35 + 254358990*x^34 + 926664900*x^33 - 2637505030*x^32 - 8389061335*x^31 + 20318079721*x^30 + 58494148730*x^29 - 117428976980*x^28 - 316667682495*x^27 + 510052425895*x^26 + 1335050206763*x^25 - 1653394093375*x^24 - 4375338382740*x^23 + 3923851568060*x^22 + 11069977015865*x^21 - 6532787291841*x^20 - 21348397264600*x^19 + 6845293868685*x^18 + 30760204370920*x^17 - 2712180162090*x^16 - 32145143355065*x^15 - 3532224571660*x^14 + 23314933370325*x^13 + 6500465979650*x^12 - 10976198292280*x^11 - 4703287953351*x^10 + 3011509345140*x^9 + 1752830160915*x^8 - 400149323895*x^7 - 327698710405*x^6 + 19849150584*x^5 + 30510229725*x^4 + 7497705*x^3 - 1362119415*x^2 - 14453695*x + 22808701)
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)]