# SageMath code for working with number field 45.45.112369590570788381690576561372238600351878236661366984431016423673684387952713968402481343001103760716137281.1

# 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 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997)

# 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 - x^44 - 132*x^43 + 301*x^42 + 7581*x^41 - 27065*x^40 - 236809*x^39 + 1229173*x^38 + 3960874*x^37 - 33065607*x^36 - 18797995*x^35 + 555716542*x^34 - 619527603*x^33 - 5732299123*x^32 + 15235581842*x^31 + 30609318626*x^30 - 168897578817*x^29 + 17112929652*x^28 + 1054352970448*x^27 - 1501837759883*x^26 - 3320354588485*x^25 + 10782132568732*x^24 - 107051646226*x^23 - 36936809485370*x^22 + 41830258905956*x^21 + 53083505529011*x^20 - 152445251342987*x^19 + 42842896041923*x^18 + 229594603485553*x^17 - 272222797945867*x^16 - 70739760279828*x^15 + 364126674087072*x^14 - 209159107972130*x^13 - 142016333800362*x^12 + 231101895832035*x^11 - 63870399263881*x^10 - 62082302112786*x^9 + 49149720219336*x^8 - 3670701881262*x^7 - 8040338248733*x^6 + 2565873490427*x^5 + 306896523050*x^4 - 230447557112*x^3 + 11171763392*x^2 + 5932497375*x - 637239997)
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)]