# SageMath code for working with number field 45.15.22655076666278607886805489601267939337377890930813594593620979789073992426604317061891.1

# 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 + 19*x^43 - 46*x^42 + 57*x^41 - 164*x^40 - 1542*x^39 + 2752*x^38 - 13177*x^37 + 27636*x^36 - 5513*x^35 + 194674*x^34 - 95004*x^33 + 1568504*x^32 - 4411428*x^31 + 7701411*x^30 - 22788169*x^29 + 356245*x^28 + 13001022*x^27 - 120681074*x^26 + 379138834*x^25 - 251876102*x^24 + 576442931*x^23 + 394617661*x^22 - 1914308187*x^21 + 1873927920*x^20 - 5326366508*x^19 + 1371219529*x^18 + 2001539759*x^17 - 9889048234*x^16 + 22330230077*x^15 - 9918095549*x^14 + 1934927850*x^13 + 41001461054*x^12 - 37979697765*x^11 + 8169529015*x^10 + 13175783912*x^9 - 55418628407*x^8 + 2188320806*x^7 - 1793697518*x^6 - 21729183123*x^5 - 927541692*x^4 + 301468299*x^3 + 1484196634*x^2 + 59977427*x - 879691)

# 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 + 19*x^43 - 46*x^42 + 57*x^41 - 164*x^40 - 1542*x^39 + 2752*x^38 - 13177*x^37 + 27636*x^36 - 5513*x^35 + 194674*x^34 - 95004*x^33 + 1568504*x^32 - 4411428*x^31 + 7701411*x^30 - 22788169*x^29 + 356245*x^28 + 13001022*x^27 - 120681074*x^26 + 379138834*x^25 - 251876102*x^24 + 576442931*x^23 + 394617661*x^22 - 1914308187*x^21 + 1873927920*x^20 - 5326366508*x^19 + 1371219529*x^18 + 2001539759*x^17 - 9889048234*x^16 + 22330230077*x^15 - 9918095549*x^14 + 1934927850*x^13 + 41001461054*x^12 - 37979697765*x^11 + 8169529015*x^10 + 13175783912*x^9 - 55418628407*x^8 + 2188320806*x^7 - 1793697518*x^6 - 21729183123*x^5 - 927541692*x^4 + 301468299*x^3 + 1484196634*x^2 + 59977427*x - 879691)
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)]