# SageMath code for working with number field 32.0.922280736601273559510361401082691485048147448883056640625.1

# 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 - 2*x^31 - 18*x^30 + 48*x^29 + 188*x^28 - 530*x^27 - 1260*x^26 + 2484*x^25 + 9755*x^24 - 11858*x^23 - 67320*x^22 + 77464*x^21 + 245214*x^20 - 151056*x^19 - 505449*x^18 - 437870*x^17 + 1660069*x^16 - 2306224*x^15 + 5411604*x^14 - 4977908*x^13 + 2763805*x^12 - 6379754*x^11 + 5902238*x^10 + 4026816*x^9 + 1973385*x^8 - 3666630*x^7 - 6023357*x^6 - 740934*x^5 - 664887*x^4 + 1401626*x^3 + 465968*x^2 + 1135240*x + 1263376)

# 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 - 2*x^31 - 18*x^30 + 48*x^29 + 188*x^28 - 530*x^27 - 1260*x^26 + 2484*x^25 + 9755*x^24 - 11858*x^23 - 67320*x^22 + 77464*x^21 + 245214*x^20 - 151056*x^19 - 505449*x^18 - 437870*x^17 + 1660069*x^16 - 2306224*x^15 + 5411604*x^14 - 4977908*x^13 + 2763805*x^12 - 6379754*x^11 + 5902238*x^10 + 4026816*x^9 + 1973385*x^8 - 3666630*x^7 - 6023357*x^6 - 740934*x^5 - 664887*x^4 + 1401626*x^3 + 465968*x^2 + 1135240*x + 1263376)
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)]