# SageMath code for working with number field 45.45.21716721949231054286553290309851361797375923410546325048215508572159529453624298126626868783164571129.1

# 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 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089)

# 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 - 12*x^44 - 50*x^43 + 1120*x^42 - 626*x^41 - 44264*x^40 + 106520*x^39 + 966733*x^38 - 3644562*x^37 - 12564205*x^36 + 68449875*x^35 + 92270622*x^34 - 830216501*x^33 - 202049055*x^32 + 6933616229*x^31 - 3181542881*x^30 - 41084005259*x^29 + 40234212427*x^28 + 174510130968*x^27 - 250382990469*x^26 - 527354195438*x^25 + 1015613064703*x^24 + 1094857977723*x^23 - 2871848912941*x^22 - 1403630755463*x^21 + 5763542313829*x^20 + 629825101852*x^19 - 8167242308412*x^18 + 1257688359124*x^17 + 7983130477206*x^16 - 2778691581109*x^15 - 5144196139934*x^14 + 2589633668016*x^13 + 2011525458240*x^12 - 1316532612469*x^11 - 396963769511*x^10 + 358975449036*x^9 + 14178066293*x^8 - 45168898819*x^7 + 5258463166*x^6 + 1537215457*x^5 - 288043322*x^4 - 5291206*x^3 + 2457345*x^2 - 9309*x - 6089)
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)]