| Curve name |
$X_{217h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \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 $X_{217h}$ minimally covers |
|
| Curves that minimally cover $X_{217h}$ |
|
| Curves that minimally cover $X_{217h}$ 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 |
$y^2 + xy = x^3 + 18913x - 381333$, with conductor $294$ |
| Generic density of odd order reductions |
$81/896$ |