# SageMath code for working with number field 24.0.304383340063522342681884765625.1 # (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^24 - x^23 + x^19 - x^18 + x^17 - x^16 + x^14 - x^13 + x^12 - x^11 + x^10 - x^8 + x^7 - x^6 + x^5 - 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)]