| Curve name |
$X_{227c}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{227}$ |
| Curves that $X_{227c}$ minimally covers |
|
| Curves that minimally cover $X_{227c}$ |
|
| Curves that minimally cover $X_{227c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -463856467968t^{32} + 13915694039040t^{30} - 15655155793920t^{28} +
6088116142080t^{26} - 1978637746176t^{24} + 271790899200t^{22} +
61152952320t^{20} - 44166021120t^{18} + 15472263168t^{16} - 2760376320t^{14} +
238878720t^{12} + 66355200t^{10} - 30191616t^{8} + 5806080t^{6} - 933120t^{4} +
51840t^{2} - 108\]
\[B(t) = 121597189939003392t^{48} + 7660622966157213696t^{46} -
31600069735398506496t^{44} + 29206125058474377216t^{42} -
15817134472534425600t^{40} + 5715542916156358656t^{38} -
1357815496442904576t^{36} + 20572962067316736t^{34} + 123565796789059584t^{32} -
60549967902670848t^{30} + 18322794341203968t^{28} - 2907684269457408t^{26} +
181730266841088t^{22} - 71573415395328t^{20} + 14782707007488t^{18} -
1885464428544t^{16} - 19619905536t^{14} + 80932110336t^{12} - 21292056576t^{10}
+ 3682713600t^{8} - 425005056t^{6} + 28740096t^{4} - 435456t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 + 981282875x + 3846528212125$, with conductor
$252150$ |
| Generic density of odd order reductions |
$51/448$ |