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| Original file line number | Diff line number | Diff line change |
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| {% extends 'homepage.html' %} | ||
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| {% block content %} | ||
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| <p> | ||
| This dataset contains all 238,764,310 {{KNOWL('ec.q', 'elliptic curves')}} over $\Q$ with {{ KNOWL('ec.q.naive_height', 'naive height') }} up to $2.7 \cdot 10^{10}$ and is described in <a href="https://doi.org/10.1112/S1461157016000152">Balakrishnan-Ho-Kaplan-Spicer-Stein-Watkins</a>. | ||
| </p> | ||
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| <h3>File and data format</h3> | ||
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| <p> | ||
| The data is stored in text files, indexed from $k=0$ to $k=2699$. The file $k$.txt contains all of the elliptic curves whose minimal short Weierstrass form $y^2 = x^3 + A x + B$ has naive height $H$ in the range: the $k$th file contains data for $k \cdot 10^7 < H \le (k+1) \cdot 10^7$. | ||
| </p> | ||
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| <p> | ||
| Each line corresponds to an elliptic curve, with columns separated by | characters and format given as follows. If a column has not been computed it is saved as the empty string. | ||
| </p> | ||
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| <table class="ntdata"> | ||
| <tr> | ||
| <th>Column(s)</th> | ||
| <th>Description</th> | ||
| <th>Example</th> | ||
| </tr> | ||
| <tr> | ||
| <td>$H$</td> | ||
| <td>{{KNOWL('ec.q.naive_height', 'Naive height')}}</td> | ||
| <td>$19003704300$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$a_1$, $a_2$, $a_3$, $a_4$, $a_6$</td> | ||
| <td>$a$-invariants for the {{KNOWL('ec.q.minimal_weierstrass_equation', 'reduced minimal model')}}</td> | ||
| <td>$1$, $-1$, $0$, $-102$, $-389$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$p_1$, $p_2$</td> | ||
| <td>Parameters to form the elliptic curve, which is isomorphic to $y^2 = x^3 + p_1 x + p_2$</td> | ||
| <td>$-1635$, $-26530$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$N$</td> | ||
| <td>{{KNOWL('ec.q.conductor', 'Conductor')}}</td> | ||
| <td>$5940675$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$\operatorname{tam}$</td> | ||
| <td>{{KNOWL('ec.tamagawa_number', 'Tamagawa product')}}</td> | ||
| <td>$1$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$n_1$, $n_2$</td> | ||
| <td>structure of {{KNOWL('ec.q.torsion_subgroup', 'torsion subgroup')}}, $\Z/n_1 \times \Z/n_2$ where $n_1 \in {1, 2}$ and $n_1 \mid n_2$</td> | ||
| <td>$1$, $1$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$\operatorname{sel}_2$</td> | ||
| <td>$2$-rank of the {{KNOWL('ag.selmer_group', '$2$-Selmer group')}}</td> | ||
| <td>$2$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$w$</td> | ||
| <td>Root number (sign of functional equation)</td> | ||
| <td>$-1$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$r_{\mathrm{an},0}$</td> | ||
| <td>Running analytic_rank_upper_bound in Sage with parameter $\delta$ (see below) returns $r_{\mathrm{an},0}$, which is an upper bound for the rank</td> | ||
| <td>$0$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$\operatorname{mw}_{\mathrm{ub}}, \operatorname{mw}_{\mathrm{lb}}, \operatorname{mw}_{\mathrm{time}}$</td> | ||
| <td>Upper and lower bounds returned by mwrank in Sage, as well as the the time used</td> | ||
| <td>$2, 2, 0.052$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$\delta$</td> | ||
| <td>Running analytic_rank_upper_bound in Sage with parameter $\delta$ returns $r_{\mathrm{an},0}$ (see above), which is an upper bound for the rank.</td> | ||
| <td>$2.0$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$\operatorname{magma}$</td> | ||
| <td>Rank computed by the Magma function MordellWeilShaInformation (almost always null)</td> | ||
| <td>$2$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$r$</td> | ||
| <td>{{KNOWL('ec.rank', 'rank')}}</td> | ||
| <td>$4$</td> | ||
| </tr> | ||
| <tr> | ||
| <td>$\operatorname{CM}$</td> | ||
| <td>1 if $E$ has {{KNOWL('ec.complex_multiplication', 'CM')}}, 0 otherwise</td> | ||
| <td>$0$</td> | ||
| </tr> | ||
| </table> | ||
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| <h3>Download</h3> | ||
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| <form> | ||
| <table> | ||
| <tr> | ||
| <td align="right">$k = $</td> | ||
| <td align="left"> | ||
| <input type="text" name="k" size="7" placeholder="123"> | ||
| <span class="formexample">integer from 0 to 2699</span> | ||
| </td> | ||
| </tr> | ||
| <tr> | ||
| <td align="left"> | ||
| <button type="submit" name="Fetch" value="fetch">Fetch file</button> | ||
| </td> | ||
| </tr> | ||
| </table> | ||
| </form> | ||
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| {# This format was nice since it gave statistics and provided direct download links, but the files are too big to download given the current 30-second timeout limitation. | ||
| <table class="ntdata"> | ||
| <tr> | ||
| <th>File</th> | ||
| <th>Size (MB)</th> | ||
| <th>Number of curves</th> | ||
| <th>Lower height bound</th> | ||
| <th>Upper height bound</th> | ||
| </tr> | ||
| {% for ncurves, size, lower, upper, fname in info.files %} | ||
| <tr> | ||
| <td><a href="{{ url_for('.render_bhkssw', filename=fname) }}">{{fname}}</a></td> | ||
| <td>{{size}}</td> | ||
| <td>{{comma(ncurves)}}</td> | ||
| <td>${{lower}}$</td> | ||
| <td>${{upper}}$</td> | ||
| </tr> | ||
| {% endfor %} | ||
| </table> | ||
| #} | ||
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| {% endblock %} |
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