# SageMath code for working with number field 45.45.40844008098536976898528926596491646392449447921859499771891413268750045487853347665634360111802937199321.1

# 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^45 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289)

# 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^45 - 3*x^44 - 126*x^43 + 350*x^42 + 7095*x^41 - 18207*x^40 - 237254*x^39 + 560949*x^38 + 5280957*x^37 - 11465163*x^36 - 83127609*x^35 + 164960568*x^34 + 959694733*x^33 - 1729668720*x^32 - 8316855543*x^31 + 13495839126*x^30 + 54911329452*x^29 - 79318322604*x^28 - 278624618803*x^27 + 353264598507*x^26 + 1090386630918*x^25 - 1193881051406*x^24 - 3286918329246*x^23 + 3054422504946*x^22 + 7584376524784*x^21 - 5885779838802*x^20 - 13239515051214*x^19 + 8489225140885*x^18 + 17170281215082*x^17 - 9118144480932*x^16 - 16115327765898*x^15 + 7281918623109*x^14 + 10535410010724*x^13 - 4323241543716*x^12 - 4518883017183*x^11 + 1877328461526*x^10 + 1138556965481*x^9 - 554295751305*x^8 - 124509763362*x^7 + 91825606123*x^6 - 3656082528*x^5 - 5240896776*x^4 + 961503906*x^3 - 1541688*x^2 - 11847522*x + 756289)
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)]