| Curve name |
$X_{213c}$ |
| 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 \\ 16 & 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 $X_{213c}$ minimally covers |
|
| Curves that minimally cover $X_{213c}$ |
|
| Curves that minimally cover $X_{213c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{22} - 270t^{20} - 783t^{18} - 648t^{16} + 378t^{14} + 972t^{12} +
378t^{10} - 648t^{8} - 783t^{6} - 270t^{4} - 27t^{2}\]
\[B(t) = 54t^{33} + 810t^{31} + 4374t^{29} + 10314t^{27} + 8910t^{25} -
6318t^{23} - 17874t^{21} - 14094t^{19} - 14094t^{17} - 17874t^{15} - 6318t^{13}
+ 8910t^{11} + 10314t^{9} + 4374t^{7} + 810t^{5} + 54t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 7824033x - 8417627937$, with conductor $196800$ |
| Generic density of odd order reductions |
$299/2688$ |