# SageMath code for working with number field 29.29.7839491297426657080705875253942679356383134413463412253884527413516798034573044321.1
# Some of these functions may take a long time to execute (this depends on the field).
# Define the number field:
x = polygen(QQ); K. = NumberField(x^29 - 406*x^27 - 261*x^26 + 60784*x^25 - 10237*x^24 - 4881280*x^23 + 6951010*x^22 + 231791925*x^21 - 664329796*x^20 - 6378435885*x^19 + 29829836682*x^18 + 83025301615*x^17 - 713976297106*x^16 + 200377868604*x^15 + 8552568793352*x^14 - 19940770675013*x^13 - 30466827556936*x^12 + 199293371073354*x^11 - 244879309451696*x^10 - 341168817490702*x^9 + 1410119390706929*x^8 - 1755729397271788*x^7 + 798559330062447*x^6 + 263826444292294*x^5 - 372309027818008*x^4 + 45256457625802*x^3 + 48917493171904*x^2 - 7142036935967*x - 2480435158303)
# 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. = NumberField(x^29 - 406*x^27 - 261*x^26 + 60784*x^25 - 10237*x^24 - 4881280*x^23 + 6951010*x^22 + 231791925*x^21 - 664329796*x^20 - 6378435885*x^19 + 29829836682*x^18 + 83025301615*x^17 - 713976297106*x^16 + 200377868604*x^15 + 8552568793352*x^14 - 19940770675013*x^13 - 30466827556936*x^12 + 199293371073354*x^11 - 244879309451696*x^10 - 341168817490702*x^9 + 1410119390706929*x^8 - 1755729397271788*x^7 + 798559330062447*x^6 + 263826444292294*x^5 - 372309027818008*x^4 + 45256457625802*x^3 + 48917493171904*x^2 - 7142036935967*x - 2480435158303)
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)]