| Curve name |
$X_{122i}$ |
| 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} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{122}$ |
| Curves that $X_{122i}$ minimally covers |
|
| Curves that minimally cover $X_{122i}$ |
|
| Curves that minimally cover $X_{122i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -110592t^{12} + 331776t^{10} - 387072t^{8} + 221184t^{6} - 62640t^{4} +
7344t^{2} - 108\]
\[B(t) = -14155776t^{18} + 63700992t^{16} - 122093568t^{14} + 130056192t^{12} -
84022272t^{10} + 33550848t^{8} - 8007552t^{6} + 1021248t^{4} - 49248t^{2} -
432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 36031x - 2644510$, with conductor $5880$ |
| Generic density of odd order reductions |
$635/5376$ |