# SageMath code for working with number field 44.0.191158492466532603992882058813386761401107535956984758058028173625115327713733756218489.1

# 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^44 - x^43 + 46*x^42 - 45*x^41 + 1195*x^40 - 1150*x^39 + 21299*x^38 - 20149*x^37 + 287730*x^36 - 267581*x^35 + 3072523*x^34 - 2804942*x^33 + 26692099*x^32 - 23887157*x^31 + 191331250*x^30 - 167444093*x^29 + 1142837707*x^28 - 975393614*x^27 + 5704151939*x^26 - 4728758325*x^25 + 23804513458*x^24 - 19075755133*x^23 + 82643603403*x^22 - 63567853467*x^21 + 237057619920*x^20 - 173490244577*x^19 + 554044068881*x^18 - 380558605544*x^17 + 1037868132567*x^16 - 657255020887*x^15 + 1512341393536*x^14 - 854306074281*x^13 + 1655465958953*x^12 - 799339188480*x^11 + 1266628178647*x^10 - 472858178519*x^9 + 634019315968*x^8 - 183040091689*x^7 + 183593805097*x^6 - 13681085952*x^5 + 19238856919*x^4 + 3193810729*x^3 + 1973987328*x^2 + 217790679*x + 27008809)

# 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^44 - x^43 + 46*x^42 - 45*x^41 + 1195*x^40 - 1150*x^39 + 21299*x^38 - 20149*x^37 + 287730*x^36 - 267581*x^35 + 3072523*x^34 - 2804942*x^33 + 26692099*x^32 - 23887157*x^31 + 191331250*x^30 - 167444093*x^29 + 1142837707*x^28 - 975393614*x^27 + 5704151939*x^26 - 4728758325*x^25 + 23804513458*x^24 - 19075755133*x^23 + 82643603403*x^22 - 63567853467*x^21 + 237057619920*x^20 - 173490244577*x^19 + 554044068881*x^18 - 380558605544*x^17 + 1037868132567*x^16 - 657255020887*x^15 + 1512341393536*x^14 - 854306074281*x^13 + 1655465958953*x^12 - 799339188480*x^11 + 1266628178647*x^10 - 472858178519*x^9 + 634019315968*x^8 - 183040091689*x^7 + 183593805097*x^6 - 13681085952*x^5 + 19238856919*x^4 + 3193810729*x^3 + 1973987328*x^2 + 217790679*x + 27008809)
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)]