# SageMath code for working with number field 29.29.13123427860740340635045684158089751314114754706664887333885491004189525894481.1

# 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 - 252*x^27 + 493*x^26 + 26301*x^25 - 70613*x^24 - 1477005*x^23 + 4757585*x^22 + 49139083*x^21 - 176708296*x^20 - 1019533247*x^19 + 3897692875*x^18 + 13650905238*x^17 - 52791707208*x^16 - 121558760223*x^15 + 441031541121*x^14 + 744433170879*x^13 - 2221117761731*x^12 - 3198643328695*x^11 + 6333895042002*x^10 + 9161654939551*x^9 - 8623697894872*x^8 - 14867816867065*x^7 + 2648025494151*x^6 + 9891108006231*x^5 + 1898151784214*x^4 - 1956687026062*x^3 - 891062263511*x^2 - 130453149826*x - 6306528127)

# 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 - 252*x^27 + 493*x^26 + 26301*x^25 - 70613*x^24 - 1477005*x^23 + 4757585*x^22 + 49139083*x^21 - 176708296*x^20 - 1019533247*x^19 + 3897692875*x^18 + 13650905238*x^17 - 52791707208*x^16 - 121558760223*x^15 + 441031541121*x^14 + 744433170879*x^13 - 2221117761731*x^12 - 3198643328695*x^11 + 6333895042002*x^10 + 9161654939551*x^9 - 8623697894872*x^8 - 14867816867065*x^7 + 2648025494151*x^6 + 9891108006231*x^5 + 1898151784214*x^4 - 1956687026062*x^3 - 891062263511*x^2 - 130453149826*x - 6306528127)
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)]