Examples

We have verified, decomposed and matched the $6{,}216{,}959$ curves over $\mathbf{Q}$ of genus $2$ in the L-functions and modular form database LMFDB.org
The algorithm verifies that the following genus 4 curve over $\mathbf{Q}(\sqrt{3})$
$$\begin{aligned} 0 &= -8x^2 + 8xy + 17y^2 - 34xz - 2yz - 28z^2 - 10xw - 9yw - 18zw + 2w^2, \\ 0 &= 4x^3 - 6x^2 y - 6x y^2 + 12x^2 z + 6xyz + 24y^2 z - 12x z^2 - 24z^3 + 2x^2 w + 7xyw \\ &\qquad + 4y^2 w + 4xzw - 13yzw - 8z^2 w - 20x w^2 - 3z w^2 - 12w^3 \end{aligned}$$
has real multiplication by the maximal order of $\mathbf{Q}(x)/(x^4 - x^3 - 3x^2 + x + 1)$.
The first step to show that, under Langlands, it corresponds to a specific Hilbert modular form $f$, i.e., $J_{\mathbf{Q}(\sqrt{3})} \sim A_f$. We used this in a recent project, where we show that the 2-isogeny field of $A_f$ solves the inverse Galois problem for $\operatorname{PSL}_2(\mathbf{F}_{16}) \rtimes C_2 \simeq \texttt{17T7}$.
Our method works just as well for isogenies and projections.
Try it: github.com/edgarcosta/endomorphisms, putatively in Magma by the end of the semester.

SOURCE: L:L281-380, compressed to one

PLANNING NOTE: Default taken. The full rigorous-endomorphism-ring slide and its day/night structure are compressed into this one. They are a method talk, and 3.2 is meant to be a shape, not a method.

PLANNING NOTE: Default taken. 3.2 runs from Leiden alone, without concrete numerical data. The UNSW 2019 deck has a 600-digit period matrix and explicit candidate matrices, but its own example stops at candidates rather than a finished certificate, and your no-new-computations rule settles it: use what exists and stop where it stops. Say the word and I will request that source instead.

Spoken: Scale. 66158 genus-2 curves from the LMFDB, verified.

Spoken: Reach. A genus-4 curve over $\mathbf{Q}(\sqrt{3})$ with RM, tied to a Hilbert modular form and to 17T7 as a Galois group over $\mathbf{Q}$.

TRANSCRIBED FROM: five-nomial-quartics/slides/mukai_leiden.tex:352-380 (the Examples frame; the Rigorous Endomorphism ring frame, 281-351, stays compressed away per the PLANNING NOTE)