# SageMath code for working with number field 36.36.240152953708250935530977810544721792914847414233751595020294189453125.1

# 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^36 - 5*x^35 - 61*x^34 + 341*x^33 + 1498*x^32 - 9923*x^31 - 18344*x^30 + 162163*x^29 + 100187*x^28 - 1651384*x^27 + 151385*x^26 + 11009448*x^25 - 6014336*x^24 - 49174374*x^23 + 42405508*x^22 + 148130790*x^21 - 162961903*x^20 - 299264731*x^19 + 387680446*x^18 + 399630129*x^17 - 587870429*x^16 - 346734395*x^15 + 566539690*x^14 + 194732689*x^13 - 340300799*x^12 - 73364329*x^11 + 122941569*x^10 + 19939192*x^9 - 25143775*x^8 - 3815281*x^7 + 2601760*x^6 + 414379*x^5 - 101999*x^4 - 16958*x^3 + 222*x^2 + 69*x + 1)

# 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^36 - 5*x^35 - 61*x^34 + 341*x^33 + 1498*x^32 - 9923*x^31 - 18344*x^30 + 162163*x^29 + 100187*x^28 - 1651384*x^27 + 151385*x^26 + 11009448*x^25 - 6014336*x^24 - 49174374*x^23 + 42405508*x^22 + 148130790*x^21 - 162961903*x^20 - 299264731*x^19 + 387680446*x^18 + 399630129*x^17 - 587870429*x^16 - 346734395*x^15 + 566539690*x^14 + 194732689*x^13 - 340300799*x^12 - 73364329*x^11 + 122941569*x^10 + 19939192*x^9 - 25143775*x^8 - 3815281*x^7 + 2601760*x^6 + 414379*x^5 - 101999*x^4 - 16958*x^3 + 222*x^2 + 69*x + 1)
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)]