| Curve name |
$X_{212c}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 16 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{212}$ |
| Curves that $X_{212c}$ minimally covers |
|
| Curves that minimally cover $X_{212c}$ |
|
| Curves that minimally cover $X_{212c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -30399297484750848t^{32} + 455989462271262720t^{30} -
256494072527585280t^{28} + 49873847435919360t^{26} - 8104500208336896t^{24} +
556627761561600t^{22} + 62620623175680t^{20} - 22613002813440t^{18} +
3960899371008t^{16} - 353328168960t^{14} + 15288238080t^{12} + 2123366400t^{10}
- 483065856t^{8} + 46448640t^{6} - 3732480t^{4} + 103680t^{2} - 108\]
\[B(t) = 2040062320599686732316672t^{48} + 64261963098890132067975168t^{46} -
132540298891460897390198784t^{44} + 61249683578629657127288832t^{42} -
16585467596672257857945600t^{40} + 2996590564425784967036928t^{38} -
355943185499528777170944t^{36} + 2696539284087339220992t^{34} +
8098008058367808897024t^{32} - 1984101348234718347264t^{30} +
300200662486285811712t^{28} - 23819749535395086336t^{26} +
372183586490548224t^{22} - 73291177364815872t^{20} + 7568745987833856t^{18} -
482678893707264t^{16} - 2511347908608t^{14} + 5179655061504t^{12} -
681345810432t^{10} + 58923417600t^{8} - 3400040448t^{6} + 114960384t^{4} -
870912t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1584300x + 1441118000$, with conductor $14400$ |
| Generic density of odd order reductions |
$51/448$ |