| Curve name |
$X_{213d}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 24 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{213}$ |
| Curves that $X_{213d}$ minimally covers |
|
| Curves that minimally cover $X_{213d}$ |
|
| Curves that minimally cover $X_{213d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{22} - 1080t^{20} - 3132t^{18} - 2592t^{16} + 1512t^{14} +
3888t^{12} + 1512t^{10} - 2592t^{8} - 3132t^{6} - 1080t^{4} - 108t^{2}\]
\[B(t) = 432t^{33} + 6480t^{31} + 34992t^{29} + 82512t^{27} + 71280t^{25} -
50544t^{23} - 142992t^{21} - 112752t^{19} - 112752t^{17} - 142992t^{15} -
50544t^{13} + 71280t^{11} + 82512t^{9} + 34992t^{7} + 6480t^{5} + 432t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - 122251x + 16425398$, with conductor $6150$ |
| Generic density of odd order reductions |
$73/672$ |