| Curve name |
$X_{102a}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{102}$ |
| Curves that $X_{102a}$ minimally covers |
|
| Curves that minimally cover $X_{102a}$ |
|
| Curves that minimally cover $X_{102a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{14} - 648t^{13} + 2484t^{12} + 11232t^{11} - 42336t^{10} -
31104t^{9} + 319680t^{8} - 518400t^{7} + 84672t^{6} + 908928t^{5} - 1619136t^{4}
+ 1451520t^{3} - 760320t^{2} + 221184t - 27648\]
\[B(t) = -432t^{21} - 3888t^{20} + 9072t^{19} + 117936t^{18} - 217728t^{17} -
1529280t^{16} + 4527360t^{15} + 6013440t^{14} - 42705792t^{13} + 53578368t^{12}
+ 69973632t^{11} - 346488192t^{10} + 659349504t^{9} - 938552832t^{8} +
1203600384t^{7} - 1358760960t^{6} + 1218945024t^{5} - 806547456t^{4} +
374685696t^{3} - 115458048t^{2} + 21233664t - 1769472\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 10804730x - 17264989353$, with conductor
$22050$ |
| Generic density of odd order reductions |
$193/1792$ |