# SageMath code for working with number field 23.23.824185149135487077883465900577270766354751717380230010246241.1

# 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^23 - 253*x^21 - 161*x^20 + 26335*x^19 + 33028*x^18 - 1467515*x^17 - 2734079*x^16 + 47658093*x^15 + 118763283*x^14 - 910024555*x^13 - 2931964974*x^12 + 9516128606*x^11 + 41347298260*x^10 - 39738537224*x^9 - 313669878607*x^8 - 104709853475*x^7 + 1076728692839*x^6 + 1230914947669*x^5 - 913297744524*x^4 - 1823221987393*x^3 - 124828056170*x^2 + 645413252986*x + 157112485811)

# 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^23 - 253*x^21 - 161*x^20 + 26335*x^19 + 33028*x^18 - 1467515*x^17 - 2734079*x^16 + 47658093*x^15 + 118763283*x^14 - 910024555*x^13 - 2931964974*x^12 + 9516128606*x^11 + 41347298260*x^10 - 39738537224*x^9 - 313669878607*x^8 - 104709853475*x^7 + 1076728692839*x^6 + 1230914947669*x^5 - 913297744524*x^4 - 1823221987393*x^3 - 124828056170*x^2 + 645413252986*x + 157112485811)
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)]