# SageMath code for working with number field 32.0.5935315803327378381589507037815252283484449810801350950039360504150390625.1

# 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^32 - x^31 + 99*x^30 + 157*x^29 + 3924*x^28 + 16492*x^27 + 103817*x^26 + 576079*x^25 + 2396881*x^24 + 11252767*x^23 + 43066027*x^22 + 154641664*x^21 + 531776968*x^20 + 1604710009*x^19 + 4592804568*x^18 + 12105474106*x^17 + 28991395516*x^16 + 65009104064*x^15 + 133385029876*x^14 + 251976154626*x^13 + 441515729309*x^12 + 705402557166*x^11 + 1044251596422*x^10 + 1417722437928*x^9 + 1747465221173*x^8 + 2010198542396*x^7 + 2043970195671*x^6 + 1742678971704*x^5 + 1178198932512*x^4 + 286794915794*x^3 - 161102822545*x^2 - 9076955370*x + 73651979561)

# 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^32 - x^31 + 99*x^30 + 157*x^29 + 3924*x^28 + 16492*x^27 + 103817*x^26 + 576079*x^25 + 2396881*x^24 + 11252767*x^23 + 43066027*x^22 + 154641664*x^21 + 531776968*x^20 + 1604710009*x^19 + 4592804568*x^18 + 12105474106*x^17 + 28991395516*x^16 + 65009104064*x^15 + 133385029876*x^14 + 251976154626*x^13 + 441515729309*x^12 + 705402557166*x^11 + 1044251596422*x^10 + 1417722437928*x^9 + 1747465221173*x^8 + 2010198542396*x^7 + 2043970195671*x^6 + 1742678971704*x^5 + 1178198932512*x^4 + 286794915794*x^3 - 161102822545*x^2 - 9076955370*x + 73651979561)
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)]