| Curve name |
$X_{219g}$ |
| 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} 1 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{219}$ |
| Curves that $X_{219g}$ minimally covers |
|
| Curves that minimally cover $X_{219g}$ |
|
| Curves that minimally cover $X_{219g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -5211t^{16} - 13392t^{15} - 16632t^{14} - 60048t^{13} - 102708t^{12} -
127440t^{11} - 289224t^{10} - 80784t^{9} - 506466t^{8} + 80784t^{7} -
289224t^{6} + 127440t^{5} - 102708t^{4} + 60048t^{3} - 16632t^{2} + 13392t -
5211\]
\[B(t) = -144774t^{24} - 558576t^{23} - 1045224t^{22} - 3299184t^{21} -
7268940t^{20} - 14034384t^{19} - 23074632t^{18} - 42888528t^{17} -
57334554t^{16} - 63764064t^{15} - 127667664t^{14} - 32169312t^{13} -
159922728t^{12} + 32169312t^{11} - 127667664t^{10} + 63764064t^{9} -
57334554t^{8} + 42888528t^{7} - 23074632t^{6} + 14034384t^{5} - 7268940t^{4} +
3299184t^{3} - 1045224t^{2} + 558576t - 144774\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 64x - 220$, with conductor $48$ |
| Generic density of odd order reductions |
$53/896$ |