# SageMath code for working with number field 44.0.1829975953789394019358992112190249409734798500798529867331745089665450995586633384881.1

# 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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416)

# 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 + 5*x^42 - 9*x^41 + 29*x^40 - 65*x^39 + 181*x^38 - 441*x^37 + 1165*x^36 - 2929*x^35 + 7589*x^34 - 19305*x^33 + 49661*x^32 - 126881*x^31 + 325525*x^30 - 833049*x^29 + 2135149*x^28 - 5467345*x^27 + 14007941*x^26 - 35877321*x^25 + 91909085*x^24 - 235418369*x^23 + 603054709*x^22 + 941673476*x^21 + 1470545360*x^20 + 2296148544*x^19 + 3586032896*x^18 + 5598561280*x^17 + 8745570304*x^16 + 13648674816*x^15 + 21333606400*x^14 + 33261092864*x^13 + 52073332736*x^12 + 80971038720*x^11 + 127322292224*x^10 + 196561862656*x^9 + 312727306240*x^8 + 473520144384*x^7 + 777389080576*x^6 + 1116691496960*x^5 + 1992864825344*x^4 + 2473901162496*x^3 + 5497558138880*x^2 + 4398046511104*x + 17592186044416)
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)]