# SageMath code for working with number field 45.45.2176994311579143098665490695095923140252535181569079054888955137870853680040351334565004592301826321.1

# 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 - x^44 - 88*x^43 + 83*x^42 + 3486*x^41 - 3092*x^40 - 82393*x^39 + 68502*x^38 + 1298642*x^37 - 1008411*x^36 - 14455343*x^35 + 10441803*x^34 + 117428375*x^33 - 78557259*x^32 - 709738502*x^31 + 437535596*x^30 + 3225742725*x^29 - 1822257498*x^28 - 11074210163*x^27 + 5697019051*x^26 + 28704796942*x^25 - 13361047180*x^24 - 55904143642*x^23 + 23423035295*x^22 + 81043444290*x^21 - 30561670072*x^20 - 86189587932*x^19 + 29605929893*x^18 + 65831554861*x^17 - 21311114591*x^16 - 35027996039*x^15 + 11388975649*x^14 + 12410963786*x^13 - 4429059287*x^12 - 2711901594*x^11 + 1176564667*x^10 + 302440276*x^9 - 189664142*x^8 - 3791931*x^7 + 14878752*x^6 - 2048263*x^5 - 256726*x^4 + 86268*x^3 - 7415*x^2 + 215*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 - x^44 - 88*x^43 + 83*x^42 + 3486*x^41 - 3092*x^40 - 82393*x^39 + 68502*x^38 + 1298642*x^37 - 1008411*x^36 - 14455343*x^35 + 10441803*x^34 + 117428375*x^33 - 78557259*x^32 - 709738502*x^31 + 437535596*x^30 + 3225742725*x^29 - 1822257498*x^28 - 11074210163*x^27 + 5697019051*x^26 + 28704796942*x^25 - 13361047180*x^24 - 55904143642*x^23 + 23423035295*x^22 + 81043444290*x^21 - 30561670072*x^20 - 86189587932*x^19 + 29605929893*x^18 + 65831554861*x^17 - 21311114591*x^16 - 35027996039*x^15 + 11388975649*x^14 + 12410963786*x^13 - 4429059287*x^12 - 2711901594*x^11 + 1176564667*x^10 + 302440276*x^9 - 189664142*x^8 - 3791931*x^7 + 14878752*x^6 - 2048263*x^5 - 256726*x^4 + 86268*x^3 - 7415*x^2 + 215*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)]