# SageMath code for working with number field 32.32.785411287106652838033363851140085365286337539393649349536153620008513161719185408.1
# Some of these functions may take a long time to execute (this depends on the field).
# Define the number field:
x = polygen(QQ); K. = NumberField(x^32 - 928*x^30 + 390224*x^28 - 98336448*x^26 + 16550375400*x^24 - 1962506736320*x^22 + 168549136238560*x^20 - 10613779893422464*x^18 + 490555639449119508*x^16 - 16494044688724018240*x^14 + 395707126668569855776*x^12 - 6557432384793443324288*x^10 + 71312077184628696151632*x^8 - 468869689667356285628544*x^6 + 1618716785756349081336640*x^4 - 2209072319385135216882944*x^2 + 500492947360694697575042)
# 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. = NumberField(x^32 - 928*x^30 + 390224*x^28 - 98336448*x^26 + 16550375400*x^24 - 1962506736320*x^22 + 168549136238560*x^20 - 10613779893422464*x^18 + 490555639449119508*x^16 - 16494044688724018240*x^14 + 395707126668569855776*x^12 - 6557432384793443324288*x^10 + 71312077184628696151632*x^8 - 468869689667356285628544*x^6 + 1618716785756349081336640*x^4 - 2209072319385135216882944*x^2 + 500492947360694697575042)
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)]