| Curve name |
$X_{108f}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 9 & 2 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 15 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{108}$ |
| Curves that $X_{108f}$ minimally covers |
|
| Curves that minimally cover $X_{108f}$ |
|
| Curves that minimally cover $X_{108f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 216t^{16} - 1728t^{15} - 22464t^{14} + 3456t^{13} + 528768t^{12} +
1278720t^{11} - 2522880t^{10} - 15496704t^{9} - 21434112t^{8} + 10976256t^{7} +
72023040t^{6} + 102795264t^{5} + 77469696t^{4} + 33951744t^{3} + 8515584t^{2} +
1105920t + 55296\]
\[B(t) = 4320t^{24} + 41472t^{23} - 103680t^{22} - 2101248t^{21} - 2467584t^{20}
+ 39979008t^{19} + 121512960t^{18} - 318504960t^{17} - 1977675264t^{16} -
693190656t^{15} + 14969069568t^{14} + 37408407552t^{13} - 1791369216t^{12} -
187519795200t^{11} - 469940797440t^{10} - 633095847936t^{9} - 525452894208t^{8}
- 251650768896t^{7} - 31558533120t^{6} + 44675629056t^{5} + 35040854016t^{4} +
13065781248t^{3} + 2824077312t^{2} + 339738624t + 17694720\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 76157x - 8065915$, with conductor $1664$ |
| Generic density of odd order reductions |
$45667/172032$ |