| Curve name |
$X_{217g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{217}$ |
| Curves that $X_{217g}$ minimally covers |
|
| Curves that minimally cover $X_{217g}$ |
|
| Curves that minimally cover $X_{217g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} + 56160t^{22} - 2862432t^{20} + 59488128t^{18} -
619949376t^{16} + 3381488640t^{14} - 9458463744t^{12} + 13525954560t^{10} -
9919190016t^{8} + 3807240192t^{6} - 732782592t^{4} + 57507840t^{2} - 442368\]
\[B(t) = -432t^{36} - 409536t^{34} + 52488000t^{32} - 2461722624t^{30} +
61032268800t^{28} - 908384440320t^{26} + 8553406353408t^{24} -
51850555785216t^{22} + 202308258521088t^{20} - 505760996818944t^{18} +
809233034084352t^{16} - 829608892563456t^{14} + 547418006618112t^{12} -
232546416721920t^{10} + 62497043251200t^{8} - 10083215867904t^{6} +
859963392000t^{4} - 26839351296t^{2} - 113246208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 1210431x + 196452927$, with conductor $9408$ |
| Generic density of odd order reductions |
$271/2688$ |