# SageMath code for working with number field 32.0.68145128239473945155058695587402151362560000000000000000.2

# 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 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256)

# 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 - 35*x^30 + 54*x^29 + 582*x^28 - 724*x^27 - 5801*x^26 + 6656*x^25 + 38405*x^24 - 49984*x^23 - 177937*x^22 + 300832*x^21 + 590114*x^20 - 1365710*x^19 - 1185291*x^18 + 4579318*x^17 + 154147*x^16 - 11942188*x^15 + 7662420*x^14 + 16564120*x^13 - 23303048*x^12 - 2384240*x^11 + 68187376*x^10 - 79122656*x^9 + 115265728*x^8 - 58417728*x^7 + 58682560*x^6 - 14078848*x^5 + 11490944*x^4 - 313088*x^3 + 444416*x^2 + 256)
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)]