# SageMath code for working with number field 42.42.1236219045653317330764639083558080790009887216550178193020957667462329030021.1 # 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^42 - 42*x^40 + 819*x^38 - 9842*x^36 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*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^42 - 42*x^40 + 819*x^38 - 9842*x^36 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*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)]