On the System of Diophantine Equations 𝑥²−6𝑦²=−5 and 𝑥=𝑎𝑧²−𝑏

Table 1

Associated number field.

𝜃^2	Integral basis	Fundamental unit	Nonassociated factor of  2−2√−5
1−√−5	1,𝜃,𝜃^2,𝜃^3	11+2𝜃−6𝜃^2−4𝜃^3	6+4𝜃−𝜃^3
2+2√−5	11,𝜃,2𝜃^2,14𝜃^3	−49−20𝜃+3𝜃^2+72𝜃^3	2+𝜃
−1−√−5	1,𝜃,𝜃^2,𝜃^3	−85+34𝜃+30𝜃^2+38𝜃^3	4+𝜃^3
−2+2√−5	11,𝜃,2𝜃^2,14𝜃^3	1−2𝜃−𝜃^2−12𝜃^3	12−𝜃