| Curve name |
$X_{92c}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{92}$ |
| Curves that $X_{92c}$ minimally covers |
|
| Curves that minimally cover $X_{92c}$ |
|
| Curves that minimally cover $X_{92c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{14} + 6696t^{12} - 27540t^{10} + 41904t^{8} - 27540t^{6} +
6696t^{4} - 108t^{2}\]
\[B(t) = 432t^{21} + 53136t^{19} - 611712t^{17} + 2274048t^{15} - 4139424t^{13}
+ 4139424t^{11} - 2274048t^{9} + 611712t^{7} - 53136t^{5} - 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 + 136120x + 66792372$, with conductor $2925$ |
| Generic density of odd order reductions |
$307/2688$ |