| Curve name |
$X_{92j}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 14 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{92}$ |
| Curves that $X_{92j}$ minimally covers |
|
| Curves that minimally cover $X_{92j}$ |
|
| Curves that minimally cover $X_{92j}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{18} + 1620t^{16} - 3564t^{14} - 1620t^{12} + 7182t^{10} -
1620t^{8} - 3564t^{6} + 1620t^{4} - 27t^{2}\]
\[B(t) = 54t^{27} + 6804t^{25} - 56376t^{23} + 74844t^{21} + 112590t^{19} -
258552t^{17} + 258552t^{13} - 112590t^{11} - 74844t^{9} + 56376t^{7} - 6804t^{5}
- 54t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 23004333x + 146811854866$, with conductor
$38025$ |
| Generic density of odd order reductions |
$307/2688$ |