| Curve name |
$X_{212e}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{212}$ |
| Curves that $X_{212e}$ minimally covers |
|
| Curves that minimally cover $X_{212e}$ |
|
| Curves that minimally cover $X_{212e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7421703487488t^{24} + 111325552312320t^{22} - 62852551409664t^{20} +
15655155793920t^{18} - 3940968038400t^{16} + 597939978240t^{14} -
92522151936t^{12} + 9342812160t^{10} - 962150400t^{8} + 59719680t^{6} -
3746304t^{4} + 103680t^{2} - 108\]
\[B(t) = -7782220156096217088t^{36} - 245139934917030838272t^{34} +
505236324196559093760t^{32} - 245139934917030838272t^{30} +
86957190455129800704t^{28} - 22742474430779228160t^{26} +
5063857976327012352t^{24} - 897729253846548480t^{22} + 144714868589592576t^{20}
- 18790361660915712t^{18} + 2261169821712384t^{16} - 219172181114880t^{14} +
19317085175808t^{12} - 1355557109760t^{10} + 80985194496t^{8} - 3567255552t^{6}
+ 114877440t^{4} - 870912t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 176033x + 53316063$, with conductor $4800$ |
| Generic density of odd order reductions |
$271/2688$ |