| Curve name |
$X_{37b}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 2 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 6 & 7 \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 $X_{37b}$ minimally covers |
|
| Curves that minimally cover $X_{37b}$ |
|
| Curves that minimally cover $X_{37b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -774144t^{8} - 2211840t^{7} - 2598912t^{6} - 1714176t^{5} - 736128t^{4}
- 214272t^{3} - 40608t^{2} - 4320t - 189\]
\[B(t) = 240648192t^{12} + 976748544t^{11} + 1677459456t^{10} + 1583677440t^{9}
+ 875225088t^{8} + 253476864t^{7} - 31684608t^{5} - 13675392t^{4} - 3093120t^{3}
- 409536t^{2} - 29808t - 918\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 66x + 230$, with conductor $640$ |
| Generic density of odd order reductions |
$1427/5376$ |