| Curve name |
$X_{120j}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{120}$ |
| Curves that $X_{120j}$ minimally covers |
|
| Curves that minimally cover $X_{120j}$ |
|
| Curves that minimally cover $X_{120j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} + 648t^{10} - 6048t^{8} + 27648t^{6} - 62640t^{4} +
58752t^{2} - 6912\]
\[B(t) = 54t^{18} - 1944t^{16} + 29808t^{14} - 254016t^{12} + 1312848t^{10} -
4193856t^{8} + 8007552t^{6} - 8169984t^{4} + 3151872t^{2} + 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 9x$, with conductor $63$ |
| Generic density of odd order reductions |
$17/168$ |