The LMFDB will soon have a new section on modular curves. And as with modular
forms, each curve will have a portrait or badge that gives a rough
approximation to some of the characteristics of the curve.

I wrote a note on some of the technical observations and implementation details
concerning these curves. This note can be found
here. I've also
added a link to it in the unpublished notes section of my research
page.

Instead of going into details here, I'll refer to the details in the note. I'll
give the core idea.

Each modular curve comes from a subgroup $H \subset \mathrm{GL}(2,
\mathbb{Z}/N\mathbb{Z})$ for some $N$ called the level. To form a
visualization, we compute cosets for $H \cap \mathrm{SL}(2,
\mathbb{Z}/N\mathbb{Z})$ inside $\mathrm{SL}(2, \mathbb{Z}/N\mathbb{Z})$, lift
these to nice elements in $\mathrm{SL}(2, \mathbb{Z})$, and then translate
the standard fundamental domain of $\mathrm{SL}(2, \mathbb{Z}) \backslash
\mathcal{H}$ by these cosets.

This is not a perfect representation, but it captures some of the character of
the curve.

Here are a few of the images that we produce.

I had studied how to produce space efficient SVG files as well, though I did
not go in this direction in the end. But I think these silhouettes are
interesting, so I include them too.

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## Info on how to comment

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won't be shared(unless you include it in the body of your comment). If you don't want your real name to be used next to your comment, please specify the name you would like to use. If you want your name to link to a particular url, include that as well.bold,italics, and plain text are allowed in comments. A reasonable subset of markdown is supported, including lists, links, and fenced code blocks. In addition, math can be formatted using`$(inline math)$`

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.Please use plaintext emailwhen commenting. See Plaintext Email and Comments on this site for more. Note also thatcomments are expected to be open, considerate, and respectful.