| Curve name |
$X_{37d}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
| $4$ |
$6$ |
$X_{9}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{37}$ |
| Curves that $X_{37d}$ minimally covers |
|
| Curves that minimally cover $X_{37d}$ |
|
| Curves that minimally cover $X_{37d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -3096576t^{8} - 8847360t^{7} - 10395648t^{6} - 6856704t^{5} -
2944512t^{4} - 857088t^{3} - 162432t^{2} - 17280t - 756\]
\[B(t) = 1925185536t^{12} + 7813988352t^{11} + 13419675648t^{10} +
12669419520t^{9} + 7001800704t^{8} + 2027814912t^{7} - 253476864t^{5} -
109403136t^{4} - 24744960t^{3} - 3276288t^{2} - 238464t - 7344\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 265x - 1575$, with conductor $640$ |
| Generic density of odd order reductions |
$401/1792$ |