# SageMath code for working with number field 45.45.23518854458506786548183561086555508726667593641132303121759124944650820067393315326853553415276110172271728515625.1

# 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 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099)

# 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 - 15*x^44 - 90*x^43 + 2245*x^42 + 840*x^41 - 149682*x^40 + 233625*x^39 + 5893830*x^38 - 15309120*x^37 - 153159405*x^36 + 505833687*x^35 + 2780167860*x^34 - 10777640365*x^33 - 36390969195*x^32 + 160894854450*x^31 + 349558305696*x^30 - 1751940770010*x^29 - 2483116182570*x^28 + 14226276458010*x^27 + 13037929556940*x^26 - 87229174471284*x^25 - 50137706272620*x^24 + 406336593221610*x^23 + 138267603359730*x^22 - 1439822630956295*x^21 - 262195938148443*x^20 + 3868718864967360*x^19 + 311910060390275*x^18 - 7821266721741750*x^17 - 175081417856730*x^16 + 11738415157749574*x^15 - 32411856380685*x^14 - 12806776910595090*x^13 + 44301971232950*x^12 + 9836514943009755*x^11 + 117791800035840*x^10 - 5063789878500180*x^9 - 176503490451285*x^8 + 1618208036238660*x^7 + 90536277582715*x^6 - 283553373081756*x^5 - 18640864878495*x^4 + 21751317924475*x^3 + 974579665770*x^2 - 434206226325*x + 15265738099)
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)]