| Curve name |
$X_{225d}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{225}$ |
| Curves that $X_{225d}$ minimally covers |
|
| Curves that minimally cover $X_{225d}$ |
|
| Curves that minimally cover $X_{225d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7421703487488t^{24} - 111325552312320t^{22} - 62388694941696t^{20} -
8697308774400t^{18} - 27179089920t^{16} + 163074539520t^{14} + 31369199616t^{12}
+ 2548039680t^{10} - 6635520t^{8} - 33177600t^{6} - 3718656t^{4} - 103680t^{2} -
108\]
\[B(t) = 7782220156096217088t^{36} - 245139934917030838272t^{34} -
505965907336193114112t^{32} - 222158066018559197184t^{30} -
39557085852032040960t^{28} - 837880636923445248t^{26} + 866973714594398208t^{24}
+ 192637735721238528t^{22} + 21154638078148608t^{20} - 330541219971072t^{16} -
47030697197568t^{14} - 3307242258432t^{12} + 49941577728t^{10} +
36840407040t^{8} + 3232825344t^{6} + 115043328t^{4} + 870912t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 1244172x + 534156784$, with conductor $2880$ |
| Generic density of odd order reductions |
$73/672$ |