| Curve name |
$X_{32a}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 9 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 12 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 12 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{32}$ |
| Curves that $X_{32a}$ minimally covers |
|
| Curves that minimally cover $X_{32a}$ |
|
| Curves that minimally cover $X_{32a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{10} - 5184t^{8} - 84672t^{6} - 497664t^{4} - 442368t^{2}\]
\[B(t) = 432t^{15} + 31104t^{13} + 881280t^{11} + 12192768t^{9} + 80953344t^{7}
+ 191102976t^{5} - 113246208t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 108075x - 13675250$, with conductor $7200$ |
| Generic density of odd order reductions |
$37/224$ |