# SageMath code for working with number field 29.29.158165571476523684791187605131317359442300677347381843010071500056545201.1

# 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 - 168*x^27 + 353*x^26 + 11480*x^25 - 34328*x^24 - 410190*x^23 + 1556443*x^22 + 8267052*x^21 - 38827232*x^20 - 94788585*x^19 + 575669857*x^18 + 574148959*x^17 - 5286405734*x^16 - 1118675686*x^15 + 30698615326*x^14 - 6935797323*x^13 - 112817035992*x^12 + 51055206761*x^11 + 256367413593*x^10 - 136472035194*x^9 - 340428884896*x^8 + 170672875682*x^7 + 236550118296*x^6 - 94550319984*x^5 - 69592403677*x^4 + 18732474566*x^3 + 3480965541*x^2 - 875365500*x + 10242329)

# 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 - 168*x^27 + 353*x^26 + 11480*x^25 - 34328*x^24 - 410190*x^23 + 1556443*x^22 + 8267052*x^21 - 38827232*x^20 - 94788585*x^19 + 575669857*x^18 + 574148959*x^17 - 5286405734*x^16 - 1118675686*x^15 + 30698615326*x^14 - 6935797323*x^13 - 112817035992*x^12 + 51055206761*x^11 + 256367413593*x^10 - 136472035194*x^9 - 340428884896*x^8 + 170672875682*x^7 + 236550118296*x^6 - 94550319984*x^5 - 69592403677*x^4 + 18732474566*x^3 + 3480965541*x^2 - 875365500*x + 10242329)
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)]