In an attempt to clarify to myself some terminology (ant the scope of the Riemann-Roch theorem), I would like to ask for examples of genus $1$ curves of the form $$C : y^2 = ax^4 + b$$ where $a, b \in \mathbb{Q}$ such that

a) the divisors of degree one at the two infinite points were not $\mathbb{Q}$-rational.

b) at least one of the divisors at the infinite points was $\mathbb{Q}$-rational.

Thank you!



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