# SageMath code for working with number field 44.0.1555282614282872031686695202841759550456839851168205668482420621934906547732043786064001.1

# 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 - x^42 - 82*x^41 + 77*x^40 + 73*x^39 + 2795*x^38 - 2452*x^37 - 2193*x^36 - 51847*x^35 + 42213*x^34 + 35713*x^33 + 576416*x^32 - 432884*x^31 - 345487*x^30 - 3995639*x^29 + 2790104*x^28 + 2010938*x^27 + 17436469*x^26 - 11948291*x^25 - 6590980*x^24 - 47156450*x^23 + 36319435*x^22 + 9637447*x^21 + 76421426*x^20 - 81395780*x^19 + 4175237*x^18 - 78460015*x^17 + 124870243*x^16 - 45435254*x^15 + 82436729*x^14 - 106904257*x^13 + 95684734*x^12 - 78566744*x^11 + 62868338*x^10 - 89632848*x^9 + 65796469*x^8 - 36457247*x^7 + 9020998*x^6 - 24902176*x^5 + 25739321*x^4 + 13163316*x^3 - 5336842*x^2 + 199578*x + 1151329)

# 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 - x^42 - 82*x^41 + 77*x^40 + 73*x^39 + 2795*x^38 - 2452*x^37 - 2193*x^36 - 51847*x^35 + 42213*x^34 + 35713*x^33 + 576416*x^32 - 432884*x^31 - 345487*x^30 - 3995639*x^29 + 2790104*x^28 + 2010938*x^27 + 17436469*x^26 - 11948291*x^25 - 6590980*x^24 - 47156450*x^23 + 36319435*x^22 + 9637447*x^21 + 76421426*x^20 - 81395780*x^19 + 4175237*x^18 - 78460015*x^17 + 124870243*x^16 - 45435254*x^15 + 82436729*x^14 - 106904257*x^13 + 95684734*x^12 - 78566744*x^11 + 62868338*x^10 - 89632848*x^9 + 65796469*x^8 - 36457247*x^7 + 9020998*x^6 - 24902176*x^5 + 25739321*x^4 + 13163316*x^3 - 5336842*x^2 + 199578*x + 1151329)
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)]