| Curve name |
$X_{40b}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 6 & 11 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 14 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{40}$ |
| Curves that $X_{40b}$ minimally covers |
|
| Curves that minimally cover $X_{40b}$ |
|
| Curves that minimally cover $X_{40b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -17280t^{4} - 6912t^{3} - 864t^{2} - 864t - 270\]
\[B(t) = 774144t^{6} + 165888t^{5} - 373248t^{4} - 235008t^{3} - 46656t^{2} +
2592t + 1512\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 35x - 49$, with conductor $1792$ |
| Generic density of odd order reductions |
$106595/344064$ |