| Curve name |
$X_{3a}$ |
| Index |
$4$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 2 & 3 \\ 3 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 3 & 3 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{3}$ |
| Curves that $X_{3a}$ minimally covers |
|
| Curves that minimally cover $X_{3a}$ |
|
| Curves that minimally cover $X_{3a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{6} + 559872t^{4} - 967458816t^{2} + 557256278016\]
\[B(t) = 432t^{9} - 2985984t^{7} + 7739670528t^{5} - 8916100448256t^{3} +
3851755393646592t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + 253x - 26862$, with conductor $1058$ |
| Generic density of odd order reductions |
$179/336$ |