| Curve name |
$X_{212h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{212}$ |
| Curves that $X_{212h}$ minimally covers |
|
| Curves that minimally cover $X_{212h}$ |
|
| Curves that minimally cover $X_{212h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -452984832t^{16} + 6794772480t^{14} - 3822059520t^{12} +
743178240t^{10} - 120987648t^{8} + 11612160t^{6} - 933120t^{4} + 25920t^{2} -
27\]
\[B(t) = 3710851743744t^{24} + 116891829927936t^{22} - 241089399226368t^{20} +
111412525400064t^{18} - 30166071902208t^{16} + 5536380616704t^{14} -
824036032512t^{12} + 86505947136t^{10} - 7364763648t^{8} + 425005056t^{6} -
14370048t^{4} + 108864t^{2} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 110x - 880$, with conductor $15$ |
| Generic density of odd order reductions |
$19/336$ |