| Curve name |
$X_{102i}$ |
| 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 & 5 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{102i}$ minimally covers |
|
| Curves that minimally cover $X_{102i}$ |
|
| Curves that minimally cover $X_{102i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{12} - 864t^{11} + 864t^{10} + 13824t^{9} - 15552t^{8} -
76032t^{7} + 183168t^{6} - 76032t^{5} - 250560t^{4} + 483840t^{3} - 400896t^{2}
+ 165888t - 27648\]
\[B(t) = 432t^{18} + 5184t^{17} + 5184t^{16} - 117504t^{15} - 145152t^{14} +
1451520t^{13} + 145152t^{12} - 10077696t^{11} + 13488768t^{10} + 17266176t^{9} -
68719104t^{8} + 102021120t^{7} - 129862656t^{6} + 174182400t^{5} -
189444096t^{4} + 138018816t^{3} - 62373888t^{2} + 15925248t - 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 1195616x + 502673184$, with conductor $5880$ |
| Generic density of odd order reductions |
$635/5376$ |