| Curve name |
$X_{193d}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 10 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{193}$ |
| Curves that $X_{193d}$ minimally covers |
|
| Curves that minimally cover $X_{193d}$ |
|
| Curves that minimally cover $X_{193d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7247757312t^{30} + 39862665216t^{28} - 84708163584t^{26} +
86520102912t^{24} - 49177165824t^{22} + 31326732288t^{20} - 25584795648t^{18} +
9884270592t^{16} - 1599049728t^{14} + 122370048t^{12} - 12006144t^{10} +
1320192t^{8} - 80784t^{6} + 2376t^{4} - 27t^{2}\]
\[B(t) = 237494511599616t^{45} - 1959329720696832t^{43} + 6857654022438912t^{41}
- 13232897318191104t^{39} + 14825780429193216t^{37} - 7780960321929216t^{35} -
3057277980377088t^{33} + 8654518015229952t^{31} - 6620674101608448t^{29} +
2627991502848000t^{27} - 694134931193856t^{25} + 173533732798464t^{23} -
41062367232000t^{21} + 6465502052352t^{19} - 528229859328t^{17} +
11662589952t^{15} + 1855125504t^{13} - 220921344t^{11} + 12324096t^{9} -
399168t^{7} + 7128t^{5} - 54t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 471879x + 73290685$, with conductor $4410$ |
| Generic density of odd order reductions |
$271/2688$ |