| Curve name |
$X_{108b}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 10 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 10 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{108}$ |
| Curves that $X_{108b}$ minimally covers |
|
| Curves that minimally cover $X_{108b}$ |
|
| Curves that minimally cover $X_{108b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 54t^{16} - 432t^{15} - 5616t^{14} + 864t^{13} + 132192t^{12} +
319680t^{11} - 630720t^{10} - 3874176t^{9} - 5358528t^{8} + 2744064t^{7} +
18005760t^{6} + 25698816t^{5} + 19367424t^{4} + 8487936t^{3} + 2128896t^{2} +
276480t + 13824\]
\[B(t) = -540t^{24} - 5184t^{23} + 12960t^{22} + 262656t^{21} + 308448t^{20} -
4997376t^{19} - 15189120t^{18} + 39813120t^{17} + 247209408t^{16} +
86648832t^{15} - 1871133696t^{14} - 4676050944t^{13} + 223921152t^{12} +
23439974400t^{11} + 58742599680t^{10} + 79136980992t^{9} + 65681611776t^{8} +
31456346112t^{7} + 3944816640t^{6} - 5584453632t^{5} - 4380106752t^{4} -
1633222656t^{3} - 353009664t^{2} - 42467328t - 2211840\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 19039x + 1017759$, with conductor $1664$ |
| Generic density of odd order reductions |
$12833/57344$ |