| Curve name |
$X_{120n}$ |
| Index |
$48$ |
| 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 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{120}$ |
| Curves that $X_{120n}$ minimally covers |
|
| Curves that minimally cover $X_{120n}$ |
|
| Curves that minimally cover $X_{120n}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{16} + 4320t^{14} - 72576t^{12} + 663552t^{10} - 3568320t^{8} +
11321856t^{6} - 19823616t^{4} + 15482880t^{2} - 1769472\]
\[B(t) = -432t^{24} + 25920t^{22} - 694656t^{20} + 10962432t^{18} -
113021568t^{16} + 797879808t^{14} - 3926264832t^{12} + 13421998080t^{10} -
31071485952t^{8} + 45951418368t^{6} - 38263062528t^{4} + 12570329088t^{2} +
905969664\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 185220230x - 970197193353$, with conductor
$22050$ |
| Generic density of odd order reductions |
$193/1792$ |