| Curve name |
$X_{121n}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{121}$ |
| Curves that $X_{121n}$ minimally covers |
|
| Curves that minimally cover $X_{121n}$ |
|
| Curves that minimally cover $X_{121n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} - 2160t^{14} - 18144t^{12} - 82944t^{10} - 223020t^{8} -
353808t^{6} - 309744t^{4} - 120960t^{2} - 6912\]
\[B(t) = -432t^{24} - 12960t^{22} - 173664t^{20} - 1370304t^{18} - 7063848t^{16}
- 24933744t^{14} - 61347888t^{12} - 104859360t^{10} - 121372992t^{8} -
89748864t^{6} - 37366272t^{4} - 6137856t^{2} + 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 13800x - 623000$, with conductor $14400$ |
| Generic density of odd order reductions |
$41/336$ |