# SageMath code for working with number field 29.1.40541810269184654077554223987906200211370142702911343138677412109288996864.1
# 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 - x^28 + 31*x^27 + 27*x^26 + 941*x^25 - 20031*x^24 - 30389*x^23 + 210983*x^22 + 9367867*x^21 + 18647003*x^20 - 51060099*x^19 - 979663711*x^18 - 741336241*x^17 - 32098060789*x^16 - 156910730311*x^15 - 132756022611*x^14 + 3952184725432*x^13 + 11271711723834*x^12 + 18506440529168*x^11 + 126537953479264*x^10 + 337098581593616*x^9 + 1206843794414080*x^8 - 3657629942420016*x^7 + 5329245181928848*x^6 - 10623419127753216*x^5 + 13803511669571360*x^4 - 12576643486615680*x^3 + 8274398085859968*x^2 - 2058625455644160*x + 214348548989952)
# 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 - x^28 + 31*x^27 + 27*x^26 + 941*x^25 - 20031*x^24 - 30389*x^23 + 210983*x^22 + 9367867*x^21 + 18647003*x^20 - 51060099*x^19 - 979663711*x^18 - 741336241*x^17 - 32098060789*x^16 - 156910730311*x^15 - 132756022611*x^14 + 3952184725432*x^13 + 11271711723834*x^12 + 18506440529168*x^11 + 126537953479264*x^10 + 337098581593616*x^9 + 1206843794414080*x^8 - 3657629942420016*x^7 + 5329245181928848*x^6 - 10623419127753216*x^5 + 13803511669571360*x^4 - 12576643486615680*x^3 + 8274398085859968*x^2 - 2058625455644160*x + 214348548989952)
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)]