\contentsline {section}{\numberline {1}Introduction}{2}{section.1}
\contentsline {subsection}{\numberline {1.1}Elliptic Curves over $\mathbb {Q}$}{2}{subsection.1.1}
\contentsline {subsection}{\numberline {1.2}Why $\mathbb {Q}(\sqrt {5})$?}{2}{subsection.1.2}
\contentsline {subsection}{\numberline {1.3}Modularity conjecture}{2}{subsection.1.3}
\contentsline {section}{\numberline {2}Computing Hilbert modular forms over $F$}{3}{section.2}
\contentsline {section}{\numberline {3}Strategies for finding an elliptic curve attached to a Hilbert modular form}{3}{section.3}
\contentsline {subsection}{\numberline {3.1}Naive enumeration}{3}{subsection.3.1}
\contentsline {subsection}{\numberline {3.2}Torsion families}{3}{subsection.3.2}
\contentsline {subsection}{\numberline {3.3}Twisting}{4}{subsection.3.3}
\contentsline {subsection}{\numberline {3.4}Curves with specified $a_\mathfrak {p}$}{4}{subsection.3.4}
\contentsline {subsection}{\numberline {3.5}Curves with good reduction outside $S$}{4}{subsection.3.5}
\contentsline {subsection}{\numberline {3.6}Special values of twisted $L$-series}{4}{subsection.3.6}
\contentsline {subsection}{\numberline {3.7}Congruence families}{4}{subsection.3.7}
\contentsline {section}{\numberline {4}Enumerating the curves in an isogeny class}{4}{section.4}
\contentsline {section}{\numberline {5}Norm conductor $31$}{4}{section.5}
\contentsline {section}{\numberline {6}Related future projects}{4}{section.6}
\contentsline {section}{\numberline {7}Tables}{4}{section.7}
\contentsline {subsection}{\numberline {7.1}Up to norm conductor $199$}{4}{subsection.7.1}
\contentsline {subsection}{\numberline {7.2}CM elliptic curves over $F$}{4}{subsection.7.2}
\contentsline {subsection}{\numberline {7.3}Extended version only: up to norm conductor $1831$}{4}{subsection.7.3}