# SageMath code for working with number field 43.43.444677695956607074780919035502815976195331208356891344496189409566167816104341684988715963441514628066825276961.1

# 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^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501)

# 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^43 - x^42 - 210*x^41 + 177*x^40 + 19424*x^39 - 12392*x^38 - 1053196*x^37 + 410572*x^36 + 37567316*x^35 - 4029555*x^34 - 936601673*x^33 - 185951871*x^32 + 16894280750*x^31 + 8734312688*x^30 - 224649876009*x^29 - 189102381409*x^28 + 2218265633870*x^27 + 2604819705111*x^26 - 16218499586789*x^25 - 24791417254787*x^24 + 86521587673786*x^23 + 168116844068603*x^22 - 325504806681996*x^21 - 819692515707425*x^20 + 795734675389219*x^19 + 2859558999201013*x^18 - 937302717934781*x^17 - 7011360351157045*x^16 - 902655412698460*x^15 + 11681897146129224*x^14 + 5604720542659268*x^13 - 12486138055465655*x^12 - 9886946345787043*x^11 + 7728522779787490*x^10 + 9075115953805640*x^9 - 2183897418754987*x^8 - 4579013283800046*x^7 - 40649514655288*x^6 + 1249295360564226*x^5 + 160575454035341*x^4 - 166814479770181*x^3 - 29312399288701*x^2 + 7715035819499*x + 1403424452501)
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)]