| Curve name |
$X_{234d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{234}$ |
| Curves that $X_{234d}$ minimally covers |
|
| Curves that minimally cover $X_{234d}$ |
|
| Curves that minimally cover $X_{234d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -124524t^{16} - 1655424t^{15} - 10219392t^{14} - 39965184t^{13} -
106106112t^{12} - 210954240t^{11} - 339462144t^{10} - 310321152t^{9} -
677873664t^{8} + 1241284608t^{7} - 5431394304t^{6} + 13501071360t^{5} -
27163164672t^{4} + 40924348416t^{3} - 41858629632t^{2} + 27122466816t -
8160804864\]
\[B(t) = -16913232t^{24} - 337271040t^{23} - 3202861824t^{22} -
19500106752t^{21} - 84816087552t^{20} - 280654295040t^{19} - 735358611456t^{18}
- 1577113141248t^{17} - 2757619593216t^{16} - 4058967048192t^{15} -
5120147718144t^{14} - 4317271031808t^{13} - 10302184488960t^{12} +
17269084127232t^{11} - 81922363490304t^{10} + 259773891084288t^{9} -
705950615863296t^{8} + 1614963856637952t^{7} - 3012028872523776t^{6} +
4598239969935360t^{5} - 5558507113807872t^{4} + 5111835984396288t^{3} -
3358444039962624t^{2} + 1414617272156160t - 283756946522112\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 1537x - 23713$, with conductor $192$ |
| Generic density of odd order reductions |
$271/2688$ |