# SageMath code for working with number field 29.29.127186199976511404062972685561977327455002805301971051425559672113022697399314124161.1

# 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 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309)

# 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 - 448*x^27 + 107*x^26 + 76138*x^25 - 43560*x^24 - 6896718*x^23 + 8492788*x^22 + 374467531*x^21 - 737137472*x^20 - 12707566814*x^19 + 34389111341*x^18 + 270515586354*x^17 - 939536322456*x^16 - 3528471156331*x^15 + 15628114907220*x^14 + 26518327124887*x^13 - 161458375653531*x^12 - 93674380846644*x^11 + 1045971567643780*x^10 - 54383106047402*x^9 - 4238937042324143*x^8 + 1725076964802557*x^7 + 10520005444822970*x^6 - 6101874655993302*x^5 - 15103434975606529*x^4 + 9152562248392343*x^3 + 10944332007104174*x^2 - 5127494290967802*x - 2711408412652309)
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)]