| Curve name |
$X_{75b}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{75}$ |
| Curves that $X_{75b}$ minimally covers |
|
| Curves that minimally cover $X_{75b}$ |
|
| Curves that minimally cover $X_{75b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{16} + 13589544960t^{14} - 3737124864t^{12} -
212336640t^{10} + 117669888t^{8} - 3317760t^{6} - 912384t^{4} + 51840t^{2} -
108\]
\[B(t) = 29686813949952t^{24} + 467567319711744t^{22} - 484266152558592t^{20} +
80363133075456t^{18} + 15111573995520t^{16} - 4337782751232t^{14} +
67777855488t^{10} - 3689349120t^{8} - 306561024t^{6} + 28864512t^{4} -
435456t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 330498271x + 793472964033$, with conductor $1138368$ |
| Generic density of odd order reductions |
$335/2688$ |