# SageMath code for working with number field 43.43.101554487341926844216969846221080781237895685553160643066505060271306560421136844000042773102578137815468250505123323781432009.1

# 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 - 462*x^41 + 301*x^40 + 94630*x^39 - 49892*x^38 - 11463758*x^37 + 6368083*x^36 + 921810336*x^35 - 636827636*x^34 - 52222121579*x^33 + 47086072273*x^32 + 2155113502777*x^31 - 2510570612339*x^30 - 65969924993323*x^29 + 96629348183472*x^28 + 1509193690295722*x^27 - 2705517116526302*x^26 - 25775660913572833*x^25 + 55430418074111539*x^24 + 325598895562519548*x^23 - 832294062829303477*x^22 - 2983438147701667253*x^21 + 9129779342869086290*x^20 + 19125104314160845818*x^19 - 72594260758345520281*x^18 - 79441906767691419891*x^17 + 413102772817597404709*x^16 + 166866840849449475306*x^15 - 1650628224018564426381*x^14 + 139804498689687778120*x^13 + 4502348802434562518439*x^12 - 1981830512581480780535*x^11 - 8037383143222291539220*x^10 + 5701308150866478483031*x^9 + 8799946070456643867879*x^8 - 8188607772821299311768*x^7 - 5304980067316890480275*x^6 + 6100770874409784770241*x^5 + 1408273844297690430983*x^4 - 2063406363097082700573*x^3 - 90731765675438007935*x^2 + 193966087122261586736*x + 18221685305593614631)

# 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 - 462*x^41 + 301*x^40 + 94630*x^39 - 49892*x^38 - 11463758*x^37 + 6368083*x^36 + 921810336*x^35 - 636827636*x^34 - 52222121579*x^33 + 47086072273*x^32 + 2155113502777*x^31 - 2510570612339*x^30 - 65969924993323*x^29 + 96629348183472*x^28 + 1509193690295722*x^27 - 2705517116526302*x^26 - 25775660913572833*x^25 + 55430418074111539*x^24 + 325598895562519548*x^23 - 832294062829303477*x^22 - 2983438147701667253*x^21 + 9129779342869086290*x^20 + 19125104314160845818*x^19 - 72594260758345520281*x^18 - 79441906767691419891*x^17 + 413102772817597404709*x^16 + 166866840849449475306*x^15 - 1650628224018564426381*x^14 + 139804498689687778120*x^13 + 4502348802434562518439*x^12 - 1981830512581480780535*x^11 - 8037383143222291539220*x^10 + 5701308150866478483031*x^9 + 8799946070456643867879*x^8 - 8188607772821299311768*x^7 - 5304980067316890480275*x^6 + 6100770874409784770241*x^5 + 1408273844297690430983*x^4 - 2063406363097082700573*x^3 - 90731765675438007935*x^2 + 193966087122261586736*x + 18221685305593614631)
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)]