| Curve name |
$X_{228h}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{228}$ |
| Curves that $X_{228h}$ minimally covers |
|
| Curves that minimally cover $X_{228h}$ |
|
| Curves that minimally cover $X_{228h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 54t^{16} + 2592t^{15} + 50112t^{14} + 518400t^{13} + 2985984t^{12} +
7133184t^{11} - 24440832t^{10} - 272056320t^{9} - 1063895040t^{8} -
2176450560t^{7} - 1564213248t^{6} + 3652190208t^{5} + 12230590464t^{4} +
16986931200t^{3} + 13136560128t^{2} + 5435817984t + 905969664\]
\[B(t) = -189t^{24} - 7776t^{23} - 111456t^{22} + 62208t^{21} + 26490240t^{20} +
470292480t^{19} + 4693248000t^{18} + 30986551296t^{17} + 138235465728t^{16} +
381505241088t^{15} + 330926653440t^{14} - 2136786075648t^{13} -
10246545211392t^{12} - 17094288605184t^{11} + 21179305820160t^{10} +
195330683437056t^{9} + 566212467621888t^{8} + 1015367312867328t^{7} +
1230306803712000t^{6} + 986274815016960t^{5} + 444432478371840t^{4} +
8349416423424t^{3} - 119674968735744t^{2} - 66795331387392t - 12987981103104\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + x$, with conductor $48$ |
| Generic density of odd order reductions |
$53/896$ |