| Curve name |
$X_{94a}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 12 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 12 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 12 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{94}$ |
| Curves that $X_{94a}$ minimally covers |
|
| Curves that minimally cover $X_{94a}$ |
|
| Curves that minimally cover $X_{94a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 81t^{20} + 3240t^{19} + 52704t^{18} + 419904t^{17} + 1166400t^{16} -
7796736t^{15} - 88915968t^{14} - 351682560t^{13} - 308551680t^{12} +
3084189696t^{11} + 14686617600t^{10} + 24673517568t^{9} - 19747307520t^{8} -
180061470720t^{7} - 364199804928t^{6} - 255483445248t^{5} + 305764761600t^{4} +
880602513408t^{3} + 884226392064t^{2} + 434865438720t + 86973087744\]
\[B(t) = 3888t^{29} + 225504t^{28} + 5906304t^{27} + 91632384t^{26} +
921922560t^{25} + 6081454080t^{24} + 23728619520t^{23} + 20145438720t^{22} -
356136321024t^{21} - 2225808211968t^{20} - 5205347794944t^{19} +
6482467749888t^{18} + 83018190422016t^{17} + 226494227939328t^{16} -
1811953823514624t^{14} - 5313164187009024t^{13} - 3319023487942656t^{12} +
21321104568090624t^{11} + 72935283489767424t^{10} + 93358999738515456t^{9} -
42248047102525440t^{8} - 398100175068856320t^{7} - 816238949553930240t^{6} -
989906811161149440t^{5} - 787116185069027328t^{4} - 405878120323743744t^{3} -
123972135054999552t^{2} - 17099604835172352t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 700700x - 620144000$, with conductor $39200$ |
| Generic density of odd order reductions |
$9249/57344$ |