| Curve name |
$X_{233f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{233}$ |
| Curves that $X_{233f}$ minimally covers |
|
| Curves that minimally cover $X_{233f}$ |
|
| Curves that minimally cover $X_{233f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1855425871872t^{24} - 60301340835840t^{22} - 192094559797248t^{20} -
249511293222912t^{18} - 162516009222144t^{16} - 55402309877760t^{14} -
9685466873856t^{12} - 865661091840t^{10} - 39676760064t^{8} - 951810048t^{6} -
11449728t^{4} - 56160t^{2} - 27\]
\[B(t) = -972777519512027136t^{36} + 57637068031087607808t^{34} +
461689330549653504000t^{32} + 1353346324723623002112t^{30} +
2097052787973449318400t^{28} + 1950740731684831887360t^{26} +
1148018767415186817024t^{24} + 434953987064309219328t^{22} +
106067792243504185344t^{20} + 16572776343763156992t^{18} +
1657309253804752896t^{16} + 106189938248122368t^{14} + 4379344052944896t^{12} +
116273208360960t^{10} + 1953032601600t^{8} + 19693780992t^{6} + 104976000t^{4} +
204768t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 128288352x + 559236806388$, with conductor $13872$ |
| Generic density of odd order reductions |
$299/2688$ |