| Curve name |
$X_{118h}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{118}$ |
| Curves that $X_{118h}$ minimally covers |
|
| Curves that minimally cover $X_{118h}$ |
|
| Curves that minimally cover $X_{118h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{14} - 864t^{12} + 13824t^{8} + 25920t^{6} - 13824t^{4} -
27648t^{2}\]
\[B(t) = 432t^{21} + 5184t^{19} + 10368t^{17} - 96768t^{15} - 445824t^{13} -
41472t^{11} + 2515968t^{9} + 3649536t^{7} + 1327104t^{5} + 1769472t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 10709868x - 13489966672$, with conductor $486720$ |
| Generic density of odd order reductions |
$307/2688$ |