| Curve name |
$X_{212b}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{212}$ |
| Curves that $X_{212b}$ minimally covers |
|
| Curves that minimally cover $X_{212b}$ |
|
| Curves that minimally cover $X_{212b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7599824371187712t^{32} + 113997365567815680t^{30} -
64123518131896320t^{28} + 12468461858979840t^{26} - 2026125052084224t^{24} +
139156940390400t^{22} + 15655155793920t^{20} - 5653250703360t^{18} +
990224842752t^{16} - 88332042240t^{14} + 3822059520t^{12} + 530841600t^{10} -
120766464t^{8} + 11612160t^{6} - 933120t^{4} + 25920t^{2} - 27\]
\[B(t) = 255007790074960841539584t^{48} + 8032745387361266508496896t^{46} -
16567537361432612173774848t^{44} + 7656210447328707140911104t^{42} -
2073183449584032232243200t^{40} + 374573820553223120879616t^{38} -
44492898187441097146368t^{36} + 337067410510917402624t^{34} +
1012251007295976112128t^{32} - 248012668529339793408t^{30} +
37525082810785726464t^{28} - 2977468691924385792t^{26} + 46522948311318528t^{22}
- 9161397170601984t^{20} + 946093248479232t^{18} - 60334861713408t^{16} -
313918488576t^{14} + 647456882688t^{12} - 85168226304t^{10} + 7365427200t^{8} -
425005056t^{6} + 14370048t^{4} - 108864t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 24755x + 2820872$, with conductor $225$ |
| Generic density of odd order reductions |
$299/2688$ |