| Curve name |
$X_{217f}$ |
| 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} 7 & 0 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{217}$ |
| Curves that $X_{217f}$ minimally covers |
|
| Curves that minimally cover $X_{217f}$ |
|
| Curves that minimally cover $X_{217f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} + 14040t^{22} - 715608t^{20} + 14872032t^{18} -
154987344t^{16} + 845372160t^{14} - 2364615936t^{12} + 3381488640t^{10} -
2479797504t^{8} + 951810048t^{6} - 183195648t^{4} + 14376960t^{2} - 110592\]
\[B(t) = -54t^{36} - 51192t^{34} + 6561000t^{32} - 307715328t^{30} +
7629033600t^{28} - 113548055040t^{26} + 1069175794176t^{24} -
6481319473152t^{22} + 25288532315136t^{20} - 63220124602368t^{18} +
101154129260544t^{16} - 103701111570432t^{14} + 68427250827264t^{12} -
29068302090240t^{10} + 7812130406400t^{8} - 1260401983488t^{6} +
107495424000t^{4} - 3354918912t^{2} - 14155776\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 302608x + 24405312$, with conductor $2352$ |
| Generic density of odd order reductions |
$299/2688$ |