# SageMath code for working with number field 32.0.2088443876129429457733048543333054873029337200425307489036314630977400864768.1

# 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 + 416*x^30 + 78416*x^28 + 8858304*x^26 + 668327400*x^24 + 35525314240*x^22 + 1367724598240*x^20 + 38608911516032*x^18 + 799928385472788*x^16 + 12056891607126080*x^14 + 129666388829365024*x^12 + 963236031303854464*x^10 + 4695775652606290512*x^8 + 13840180870839593088*x^6 + 21419327538204132160*x^4 + 13103588611607233792*x^2 + 1330833218366359682)

# 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 + 416*x^30 + 78416*x^28 + 8858304*x^26 + 668327400*x^24 + 35525314240*x^22 + 1367724598240*x^20 + 38608911516032*x^18 + 799928385472788*x^16 + 12056891607126080*x^14 + 129666388829365024*x^12 + 963236031303854464*x^10 + 4695775652606290512*x^8 + 13840180870839593088*x^6 + 21419327538204132160*x^4 + 13103588611607233792*x^2 + 1330833218366359682)
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)]