# SageMath code for working with number field 32.32.2318482735674757180308401186646582923236113207344277714859125196933746337890625.1

# 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 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1028778*x^25 + 17622874*x^24 - 28355437*x^23 - 306303044*x^22 + 508700394*x^21 + 3552738857*x^20 - 6168097580*x^19 - 27704655126*x^18 + 51265014015*x^17 + 143975452663*x^16 - 291596165966*x^15 - 484526195470*x^14 + 1120910038107*x^13 + 989059812617*x^12 - 2853145220348*x^11 - 1010392771187*x^10 + 4674809722638*x^9 - 46102330825*x^8 - 4692807095098*x^7 + 1274517894230*x^6 + 2597013247154*x^5 - 1244875523883*x^4 - 588593534703*x^3 + 420651733214*x^2 - 11758954445*x - 18277845959)

# 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 - 192*x^30 + 254*x^29 + 15176*x^28 - 22502*x^27 - 659185*x^26 + 1028778*x^25 + 17622874*x^24 - 28355437*x^23 - 306303044*x^22 + 508700394*x^21 + 3552738857*x^20 - 6168097580*x^19 - 27704655126*x^18 + 51265014015*x^17 + 143975452663*x^16 - 291596165966*x^15 - 484526195470*x^14 + 1120910038107*x^13 + 989059812617*x^12 - 2853145220348*x^11 - 1010392771187*x^10 + 4674809722638*x^9 - 46102330825*x^8 - 4692807095098*x^7 + 1274517894230*x^6 + 2597013247154*x^5 - 1244875523883*x^4 - 588593534703*x^3 + 420651733214*x^2 - 11758954445*x - 18277845959)
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)]