| Curve name |
$X_{227e}$ |
| Index |
$96$ |
| Level |
$16$ |
| 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 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{227}$ |
| Curves that $X_{227e}$ minimally covers |
|
| Curves that minimally cover $X_{227e}$ |
|
| Curves that minimally cover $X_{227e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{24} + 54358179840t^{22} - 61379444736t^{20} +
30576476160t^{18} - 15394406400t^{16} + 4671406080t^{14} - 1445658624t^{12} +
291962880t^{10} - 60134400t^{8} + 7464960t^{6} - 936576t^{4} + 51840t^{2} -
108\]
\[B(t) = -29686813949952t^{36} - 1870269278846976t^{34} + 7709294497628160t^{32}
- 7481077115387904t^{30} + 5307445706489856t^{28} - 2776180960788480t^{26} +
1236293451251712t^{24} - 438344362229760t^{22} + 141323113857024t^{20} -
36699925118976t^{18} + 8832694616064t^{16} - 1712282664960t^{14} +
301829455872t^{12} - 42361159680t^{10} + 5061574656t^{8} - 445906944t^{6} +
28719360t^{4} - 435456t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + 39251315x + 30772225697$, with conductor $50430$ |
| Generic density of odd order reductions |
$11/112$ |