| Curve name |
$X_{102k}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{102k}$ minimally covers |
|
| Curves that minimally cover $X_{102k}$ |
|
| Curves that minimally cover $X_{102k}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{12} - 216t^{11} + 216t^{10} + 3456t^{9} - 3888t^{8} - 19008t^{7}
+ 45792t^{6} - 19008t^{5} - 62640t^{4} + 120960t^{3} - 100224t^{2} + 41472t -
6912\]
\[B(t) = 54t^{18} + 648t^{17} + 648t^{16} - 14688t^{15} - 18144t^{14} +
181440t^{13} + 18144t^{12} - 1259712t^{11} + 1686096t^{10} + 2158272t^{9} -
8589888t^{8} + 12752640t^{7} - 16232832t^{6} + 21772800t^{5} - 23680512t^{4} +
17252352t^{3} - 7796736t^{2} + 1990656t - 221184\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 30592x + 2036852$, with conductor $2352$ |
| Generic density of odd order reductions |
$17/168$ |