# SageMath code for working with number field 42.42.5239206534209069133889646882729090965733830111939046184442828436267483950448112600301.1 # 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 - x^41 - 85*x^40 + 85*x^39 + 3355*x^38 - 3355*x^37 - 81613*x^36 + 81613*x^35 + 1369379*x^34 - 1369379*x^33 - 16806205*x^32 + 16806205*x^31 + 156107459*x^30 - 156107459*x^29 - 1120160061*x^28 + 1120160061*x^27 + 6282191555*x^26 - 6282191555*x^25 - 27681539389*x^24 + 27681539389*x^23 + 95822936771*x^22 - 95822936771*x^21 - 259252432189*x^20 + 259252432189*x^19 + 542530659011*x^18 - 542530659011*x^17 - 863673531709*x^16 + 863673531709*x^15 + 1020501541571*x^14 - 1020501541571*x^13 - 863673531709*x^12 + 863673531709*x^11 + 497119576771*x^10 - 497119576771*x^9 - 180198260029*x^8 + 180198260029*x^7 + 36543447747*x^6 - 36543447747*x^5 - 3382656317*x^4 + 3382656317*x^3 + 89178819*x^2 - 89178819*x - 998717) # 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 - x^41 - 85*x^40 + 85*x^39 + 3355*x^38 - 3355*x^37 - 81613*x^36 + 81613*x^35 + 1369379*x^34 - 1369379*x^33 - 16806205*x^32 + 16806205*x^31 + 156107459*x^30 - 156107459*x^29 - 1120160061*x^28 + 1120160061*x^27 + 6282191555*x^26 - 6282191555*x^25 - 27681539389*x^24 + 27681539389*x^23 + 95822936771*x^22 - 95822936771*x^21 - 259252432189*x^20 + 259252432189*x^19 + 542530659011*x^18 - 542530659011*x^17 - 863673531709*x^16 + 863673531709*x^15 + 1020501541571*x^14 - 1020501541571*x^13 - 863673531709*x^12 + 863673531709*x^11 + 497119576771*x^10 - 497119576771*x^9 - 180198260029*x^8 + 180198260029*x^7 + 36543447747*x^6 - 36543447747*x^5 - 3382656317*x^4 + 3382656317*x^3 + 89178819*x^2 - 89178819*x - 998717) 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)]