# SageMath code for working with number field 31.31.10780058321499447579019490584061143592974328542534512659762928215062178274195635731849.1

# 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 - 330*x^29 + 852*x^28 + 43784*x^27 - 173944*x^26 - 2923572*x^25 + 15562316*x^24 + 100635650*x^23 - 707709554*x^22 - 1616621784*x^21 + 17266725972*x^20 + 7046247776*x^19 - 243865756432*x^18 + 142454592564*x^17 + 2084163747748*x^16 - 2521472241195*x^15 - 10836712986085*x^14 + 18499228115810*x^13 + 32698257976552*x^12 - 74538290301632*x^11 - 47884603550112*x^10 + 168256221920320*x^9 + 4878741111552*x^8 - 195294436805376*x^7 + 62882730299136*x^6 + 92885896276480*x^5 - 43440624908288*x^4 - 15171034030080*x^3 + 6823763935232*x^2 + 786648006656*x + 5192548352)

# 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 - 330*x^29 + 852*x^28 + 43784*x^27 - 173944*x^26 - 2923572*x^25 + 15562316*x^24 + 100635650*x^23 - 707709554*x^22 - 1616621784*x^21 + 17266725972*x^20 + 7046247776*x^19 - 243865756432*x^18 + 142454592564*x^17 + 2084163747748*x^16 - 2521472241195*x^15 - 10836712986085*x^14 + 18499228115810*x^13 + 32698257976552*x^12 - 74538290301632*x^11 - 47884603550112*x^10 + 168256221920320*x^9 + 4878741111552*x^8 - 195294436805376*x^7 + 62882730299136*x^6 + 92885896276480*x^5 - 43440624908288*x^4 - 15171034030080*x^3 + 6823763935232*x^2 + 786648006656*x + 5192548352)
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)]