# SageMath code for working with number field 28.28.2070706293589565601613551437543564286910572644210741.1 # 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^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309) # 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^28 - x^27 - 57*x^26 + 57*x^25 + 1451*x^24 - 1451*x^23 - 21749*x^22 + 21749*x^21 + 213035*x^20 - 213035*x^19 - 1430453*x^18 + 1430453*x^17 + 6715531*x^16 - 6715531*x^15 - 22059893*x^14 + 22059893*x^13 + 49878667*x^12 - 49878667*x^11 - 74814837*x^10 + 74814837*x^9 + 69567115*x^8 - 69567115*x^7 - 35437941*x^6 + 35437941*x^5 + 7799435*x^4 - 7799435*x^3 - 515445*x^2 + 515445*x - 40309) 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)]