# SageMath code for working with number field 35.35.876564456148583685580741416193317498080031692578600918378223281.1 # 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^35 - x^34 - 34*x^33 + 33*x^32 + 528*x^31 - 496*x^30 - 4960*x^29 + 4495*x^28 + 31465*x^27 - 27405*x^26 - 142506*x^25 + 118755*x^24 + 475020*x^23 - 376740*x^22 - 1184040*x^21 + 888030*x^20 + 2220075*x^19 - 1562275*x^18 - 3124550*x^17 + 2042975*x^16 + 3268760*x^15 - 1961256*x^14 - 2496144*x^13 + 1352078*x^12 + 1352078*x^11 - 646646*x^10 - 497420*x^9 + 203490*x^8 + 116280*x^7 - 38760*x^6 - 15504*x^5 + 3876*x^4 + 969*x^3 - 153*x^2 - 18*x + 1) # 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^35 - x^34 - 34*x^33 + 33*x^32 + 528*x^31 - 496*x^30 - 4960*x^29 + 4495*x^28 + 31465*x^27 - 27405*x^26 - 142506*x^25 + 118755*x^24 + 475020*x^23 - 376740*x^22 - 1184040*x^21 + 888030*x^20 + 2220075*x^19 - 1562275*x^18 - 3124550*x^17 + 2042975*x^16 + 3268760*x^15 - 1961256*x^14 - 2496144*x^13 + 1352078*x^12 + 1352078*x^11 - 646646*x^10 - 497420*x^9 + 203490*x^8 + 116280*x^7 - 38760*x^6 - 15504*x^5 + 3876*x^4 + 969*x^3 - 153*x^2 - 18*x + 1) 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)]