| Curve name |
$X_{215a}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{215}$ |
| Curves that $X_{215a}$ minimally covers |
|
| Curves that minimally cover $X_{215a}$ |
|
| Curves that minimally cover $X_{215a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} - 864t^{20} + 13824t^{12} - 221184t^{4} - 442368\]
\[B(t) = 432t^{36} + 5184t^{32} + 10368t^{28} - 96768t^{24} - 663552t^{20} -
2654208t^{16} - 6193152t^{12} + 10616832t^{8} + 84934656t^{4} + 113246208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 8033x + 275937$, with conductor $4800$ |
| Generic density of odd order reductions |
$109/896$ |