| Curve name |
$X_{228g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{228}$ |
| Curves that $X_{228g}$ minimally covers |
|
| Curves that minimally cover $X_{228g}$ |
|
| Curves that minimally cover $X_{228g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 216t^{16} + 10368t^{15} + 200448t^{14} + 2073600t^{13} + 11943936t^{12}
+ 28532736t^{11} - 97763328t^{10} - 1088225280t^{9} - 4255580160t^{8} -
8705802240t^{7} - 6256852992t^{6} + 14608760832t^{5} + 48922361856t^{4} +
67947724800t^{3} + 52546240512t^{2} + 21743271936t + 3623878656\]
\[B(t) = 1512t^{24} + 62208t^{23} + 891648t^{22} - 497664t^{21} -
211921920t^{20} - 3762339840t^{19} - 37545984000t^{18} - 247892410368t^{17} -
1105883725824t^{16} - 3052041928704t^{15} - 2647413227520t^{14} +
17094288605184t^{13} + 81972361691136t^{12} + 136754308841472t^{11} -
169434446561280t^{10} - 1562645467496448t^{9} - 4529699740975104t^{8} -
8122938502938624t^{7} - 9842454429696000t^{6} - 7890198520135680t^{5} -
3555459826974720t^{4} - 66795331387392t^{3} + 957399749885952t^{2} +
534362651099136t + 103903848824832\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 3x + 3$, with conductor $192$ |
| Generic density of odd order reductions |
$271/2688$ |