Portraits of Number Fields

Håvard Damm-Johnsen
University of Gothenburg

### Portraits in the LMFDB
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Many objects in the lmfdb have portraits

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Ideally, should be generated on the fly

_What does a number field look like?_

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## Spectrum of a ring
Let $R$ be a ring.

$\operatorname{Spec} R = \\{ \text{ prime ideals } \mathfrak p \leq R  \\}$

e.g. $\operatorname{Spec} \mathbb Z = \\{(0), (2), (3), (5) \ldots\\} $

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## $\operatorname{Spec} \mathbb Z$

<div style="display: inline-grid; width: 100%; justify-items: center;">
      <img style="width: 50%; grid-area: 1 / 1" src="/lucant25/specZ_noline.svg" alt="Spectrum of the integers">
      <img style="width: 50%; grid-area: 1 / 1"" src="/lucant25/specZ.svg" alt="Spectrum of the integers">
  </div>
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## $\operatorname{Spec} \mathbb Z [i]$

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## Inspiration: Mumford's $\operatorname{Spec} \mathbb Z [x]$

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## [Drawspec v1 (2023)](https://github.com/havarddj/drawspec/)
-  Draws $\operatorname{Spec} \mathcal{O}_F$ for any number field $F$<div data-markdown>
-  Combines [`pytikz`](https://github.com/allefeld/pytikz), a python interface to `tikz`, with `sage`.
-  <a target="_blank" rel="noopener noreferrer" href="https://github.com/havarddj/drawspec/blob/main/Drawing%20spectra%20of%20number%20fields.ipynb"> Jupyter notebook </a>

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  ### Another example of `pytikz`:
<img style="width: 60%" src="/lucant25/X0(p).png">

(~40 loc, from the <a href="https://ems.press/books/mlm/297">Eigenvarieties book </a>)
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## Problem: Can't compile tikz in the browser!

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## [Drawspec v2 (2025)](https://github.com/LMFDB/lmfdb/blob/main/lmfdb/number_fields/draw_spectrum.py)
-  Uses [`svg.py`](https://github.com/orsinium-labs/svg.py), a python library to generate SVG files
- SVG = Scalable Vector Graphics - vectorized images

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<img src="/lucant25/drawspec2_spec.svg">

- Color corresponds to ramification index
- Size corresponds to degree of residue field

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## Picture in the properties box?

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## <a href="https://www.lmfdb.org/NumberField/7.1.3911915096945863.1"> Spectra in the LMFDB </a>

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## Thanks to:
  - The LMFDB team for helpful discussion
  - James Taylor & Jay Swar for brainstorming