# SageMath code for working with number field 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1 # 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^44 - x^43 + 11*x^42 - 12*x^41 + 77*x^40 - 74*x^39 + 424*x^38 - 368*x^37 + 2052*x^36 - 1675*x^35 + 8154*x^34 - 6259*x^33 + 27860*x^32 - 18614*x^31 + 82991*x^30 - 50150*x^29 + 218995*x^28 - 122605*x^27 + 487495*x^26 - 239465*x^25 + 942321*x^24 - 367121*x^23 + 1562162*x^22 - 556097*x^21 + 2230394*x^20 - 742219*x^19 + 2512113*x^18 - 607127*x^17 + 2369760*x^16 - 263798*x^15 + 1750478*x^14 - 273124*x^13 + 1076097*x^12 - 237409*x^11 + 402111*x^10 - 41911*x^9 + 123922*x^8 + 19642*x^7 + 17600*x^6 + 1439*x^5 + 2686*x^4 - 361*x^3 + 51*x^2 - 6*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^44 - x^43 + 11*x^42 - 12*x^41 + 77*x^40 - 74*x^39 + 424*x^38 - 368*x^37 + 2052*x^36 - 1675*x^35 + 8154*x^34 - 6259*x^33 + 27860*x^32 - 18614*x^31 + 82991*x^30 - 50150*x^29 + 218995*x^28 - 122605*x^27 + 487495*x^26 - 239465*x^25 + 942321*x^24 - 367121*x^23 + 1562162*x^22 - 556097*x^21 + 2230394*x^20 - 742219*x^19 + 2512113*x^18 - 607127*x^17 + 2369760*x^16 - 263798*x^15 + 1750478*x^14 - 273124*x^13 + 1076097*x^12 - 237409*x^11 + 402111*x^10 - 41911*x^9 + 123922*x^8 + 19642*x^7 + 17600*x^6 + 1439*x^5 + 2686*x^4 - 361*x^3 + 51*x^2 - 6*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)]