# SageMath code for working with number field 29.1.142449251725173555024565249558533387292386826365409.1

# 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^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375)

# 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^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375)
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)]