| Curve name |
$X_{219d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{219}$ |
| Curves that $X_{219d}$ minimally covers |
|
| Curves that minimally cover $X_{219d}$ |
|
| Curves that minimally cover $X_{219d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -20844t^{16} - 53568t^{15} - 66528t^{14} - 240192t^{13} - 410832t^{12}
- 509760t^{11} - 1156896t^{10} - 323136t^{9} - 2025864t^{8} + 323136t^{7} -
1156896t^{6} + 509760t^{5} - 410832t^{4} + 240192t^{3} - 66528t^{2} + 53568t -
20844\]
\[B(t) = -1158192t^{24} - 4468608t^{23} - 8361792t^{22} - 26393472t^{21} -
58151520t^{20} - 112275072t^{19} - 184597056t^{18} - 343108224t^{17} -
458676432t^{16} - 510112512t^{15} - 1021341312t^{14} - 257354496t^{13} -
1279381824t^{12} + 257354496t^{11} - 1021341312t^{10} + 510112512t^{9} -
458676432t^{8} + 343108224t^{7} - 184597056t^{6} + 112275072t^{5} -
58151520t^{4} + 26393472t^{3} - 8361792t^{2} + 4468608t - 1158192\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 257x - 1503$, with conductor $192$ |
| Generic density of odd order reductions |
$109/896$ |