I am reading Poonen's paper The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture. Several proofs require recognizing certain hyperelliptic curves are actually modular curves, for example,
- $y^2 = x^6 + 2x^5 + 5x^4 + 10x^3 + 10x^2 + 4x + 1$
- $y^2 = x^6 + 2x^5 + x^4 + 2x^3 + 6x^2 + 4x + 1$
Question In general how do we know these are modular curves/map it back to some well known curves?
Poonen has a remark
The curves were originally recognized as $X_1(13)$ and $X_1(18)$ by computing enough invariants (such as the genus, automorphism group, primes of bad reduction, and Mordell-Weil group of the Jacobian) that the result could be guessed.
It seems like some luck is needed here. Since the paper was written almost 30 years ago, is our approach today still the same?
Thank you.