| Curve name |
$X_{203g}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{203}$ |
| Curves that $X_{203g}$ minimally covers |
|
| Curves that minimally cover $X_{203g}$ |
|
| Curves that minimally cover $X_{203g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7421703487488t^{24} - 18554258718720t^{22} - 44762149158912t^{20} -
62214748766208t^{18} - 41252422680576t^{16} - 13634843443200t^{14} -
2457782452224t^{12} - 213044428800t^{10} - 10071392256t^{8} - 237330432t^{6} -
2668032t^{4} - 17280t^{2} - 108\]
\[B(t) = 7782220156096217088t^{36} + 29183325585360814080t^{34} -
16415620641765457920t^{32} - 163183428898142552064t^{30} -
271610123201877639168t^{28} - 244940439527287160832t^{26} -
142280943195981348864t^{24} - 54465625696814235648t^{22} -
13305326650238435328t^{20} - 2060618114827026432t^{18} -
207895728909975552t^{16} - 13297271898636288t^{14} - 542758724960256t^{12} -
14599587889152t^{10} - 252956639232t^{8} - 2374631424t^{6} - 3732480t^{4} +
103680t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 32072449x - 69918983329$, with conductor $55488$ |
| Generic density of odd order reductions |
$109/896$ |