| Curve name |
$X_{79e}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 4 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 12 & 11 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{79}$ |
| Curves that $X_{79e}$ minimally covers |
|
| Curves that minimally cover $X_{79e}$ |
|
| Curves that minimally cover $X_{79e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -891t^{20} - 14580t^{19} - 102816t^{18} - 402408t^{17} - 899100t^{16} -
844992t^{15} + 1130112t^{14} + 4613760t^{13} + 4937760t^{12} - 2395008t^{11} -
11059200t^{10} - 4790016t^{9} + 19751040t^{8} + 36910080t^{7} + 18081792t^{6} -
27039744t^{5} - 57542400t^{4} - 51508224t^{3} - 26320896t^{2} - 7464960t -
912384\]
\[B(t) = 10206t^{30} + 249804t^{29} + 2767284t^{28} + 18179856t^{27} +
76941576t^{26} + 205274736t^{25} + 252226224t^{24} - 462765312t^{23} -
3127592736t^{22} - 7745626944t^{21} - 9546330816t^{20} + 2319881472t^{19} +
36116316288t^{18} + 78452064000t^{17} + 83647924992t^{16} - 167295849984t^{14} -
313808256000t^{13} - 288930530304t^{12} - 37118103552t^{11} + 305482586112t^{10}
+ 495720124416t^{9} + 400331870208t^{8} + 118467919872t^{7} - 129139826688t^{6}
- 210201329664t^{5} - 157576347648t^{4} - 74464690176t^{3} - 22669590528t^{2} -
4092788736t - 334430208\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 16007075x - 24649952250$, with conductor $39200$ |
| Generic density of odd order reductions |
$9249/57344$ |