# SageMath code for working with number field 31.31.606473279060866185803408353837168008326030742006308575695635765007438922801.1

# 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^31 - x^30 - 150*x^29 + 117*x^28 + 9434*x^27 - 4958*x^26 - 328880*x^25 + 78328*x^24 + 7098810*x^23 + 507925*x^22 - 100297691*x^21 - 39217806*x^20 + 950942678*x^19 + 677567200*x^18 - 6056859921*x^17 - 6293114670*x^16 + 25303767350*x^15 + 35315191500*x^14 - 65145555320*x^13 - 122001770825*x^12 + 87706314807*x^11 + 253609179978*x^10 - 19776440614*x^9 - 299693882286*x^8 - 98200795275*x^7 + 175359843704*x^6 + 113820457794*x^5 - 30260338943*x^4 - 38729335872*x^3 - 5811762482*x^2 + 1056847214*x - 26458109)

# 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^31 - x^30 - 150*x^29 + 117*x^28 + 9434*x^27 - 4958*x^26 - 328880*x^25 + 78328*x^24 + 7098810*x^23 + 507925*x^22 - 100297691*x^21 - 39217806*x^20 + 950942678*x^19 + 677567200*x^18 - 6056859921*x^17 - 6293114670*x^16 + 25303767350*x^15 + 35315191500*x^14 - 65145555320*x^13 - 122001770825*x^12 + 87706314807*x^11 + 253609179978*x^10 - 19776440614*x^9 - 299693882286*x^8 - 98200795275*x^7 + 175359843704*x^6 + 113820457794*x^5 - 30260338943*x^4 - 38729335872*x^3 - 5811762482*x^2 + 1056847214*x - 26458109)
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)]