Abstract
<p> We present results of quadratic Chabauty experiments on genus 2 bielliptic modular curves of Jacobian rank 2 that have recently been added to the LMFDB. We apply quadratic Chabauty methods over both the rationals and quadratic imaginary fields. In a number of cases, the experiments yielded algebraic irrational points among the set of mock rational points. We highlight specific notable examples, including the non-split Cartan modular curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Subscript n s Superscript plus Baseline left-parenthesis 15 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mn>15</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">X_{ns}^+(15)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Lastly, we offer a conjecture relating the level of the modular curve to the potential number fields over which points can arise. </p>