# SageMath code for working with number field 32.0.2553263220825544945190771147906377486172160000000000000000.2 # 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^32 - 2*x^31 - 39*x^30 + 78*x^29 + 678*x^28 - 1336*x^27 - 6933*x^26 + 13256*x^25 + 46161*x^24 - 84484*x^23 - 207785*x^22 + 363100*x^21 + 629290*x^20 - 1071214*x^19 - 1236459*x^18 + 2000398*x^17 + 1905251*x^16 - 877660*x^15 - 6200876*x^14 - 2748360*x^13 + 11687032*x^12 - 12604592*x^11 + 40542960*x^10 - 4541024*x^9 + 55107776*x^8 + 1846464*x^7 + 23026880*x^6 + 1271424*x^5 + 3919488*x^4 + 158976*x^3 + 231424*x^2 + 256) # 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^32 - 2*x^31 - 39*x^30 + 78*x^29 + 678*x^28 - 1336*x^27 - 6933*x^26 + 13256*x^25 + 46161*x^24 - 84484*x^23 - 207785*x^22 + 363100*x^21 + 629290*x^20 - 1071214*x^19 - 1236459*x^18 + 2000398*x^17 + 1905251*x^16 - 877660*x^15 - 6200876*x^14 - 2748360*x^13 + 11687032*x^12 - 12604592*x^11 + 40542960*x^10 - 4541024*x^9 + 55107776*x^8 + 1846464*x^7 + 23026880*x^6 + 1271424*x^5 + 3919488*x^4 + 158976*x^3 + 231424*x^2 + 256) 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)]