# SageMath code for working with number field 17.1.463009808974713123841.1 # (Note that not all these functions may be available, and some may take a long time to execute.) # Define the number field: x = polygen(QQ); K. = NumberField(x^17 - x^16 - x^15 - x^14 + x^12 + 13*x^11 + 7*x^10 + 11*x^9 + 4*x^8 + x^7 + 7*x^6 + 23*x^5 + 31*x^4 + 42*x^3 + 24*x^2 + 6*x - 1) # Defining polynomial: K.defining_polynomial() # Degree over Q: K.degree() # Signature: K.signature() # Discriminant: K.disc() # Ramified primes: K.disc().support() # 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() # Galois group: K.galois_group(type='pari') # Frobenius cycle types: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]