# SageMath code for working with number field 32.0.170615961014359235699341036323165930807590484619140625.1.
# 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 - 9*x^31 + 200*x^29 - 221*x^28 - 2631*x^27 + 4402*x^26 + 19766*x^25 - 27449*x^24 - 173128*x^23 + 338537*x^22 + 473336*x^21 - 1004401*x^20 - 3368948*x^19 + 11453717*x^18 - 8139429*x^17 - 16834230*x^16 - 597015*x^15 + 195460786*x^14 - 382818671*x^13 - 27697040*x^12 + 420160443*x^11 + 1322753872*x^10 - 3917503795*x^9 + 2111159122*x^8 + 2050419221*x^7 + 1497324960*x^6 - 10922257252*x^5 + 11658561303*x^4 - 2839464368*x^3 - 2153549419*x^2 + 711827200*x + 313809121)

# 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 - 9*x^31 + 200*x^29 - 221*x^28 - 2631*x^27 + 4402*x^26 + 19766*x^25 - 27449*x^24 - 173128*x^23 + 338537*x^22 + 473336*x^21 - 1004401*x^20 - 3368948*x^19 + 11453717*x^18 - 8139429*x^17 - 16834230*x^16 - 597015*x^15 + 195460786*x^14 - 382818671*x^13 - 27697040*x^12 + 420160443*x^11 + 1322753872*x^10 - 3917503795*x^9 + 2111159122*x^8 + 2050419221*x^7 + 1497324960*x^6 - 10922257252*x^5 + 11658561303*x^4 - 2839464368*x^3 - 2153549419*x^2 + 711827200*x + 313809121)
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 pO_K for p=7 in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]