| Curve name |
$X_{203c}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 2 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{203}$ |
| Curves that $X_{203c}$ minimally covers |
|
| Curves that minimally cover $X_{203c}$ |
|
| Curves that minimally cover $X_{203c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1855425871872t^{24} - 4638564679680t^{22} - 11190537289728t^{20} -
15553687191552t^{18} - 10313105670144t^{16} - 3408710860800t^{14} -
614445613056t^{12} - 53261107200t^{10} - 2517848064t^{8} - 59332608t^{6} -
667008t^{4} - 4320t^{2} - 27\]
\[B(t) = -972777519512027136t^{36} - 3647915698170101760t^{34} +
2051952580220682240t^{32} + 20397928612267819008t^{30} +
33951265400234704896t^{28} + 30617554940910895104t^{26} +
17785117899497668608t^{24} + 6808203212101779456t^{22} +
1663165831279804416t^{20} + 257577264353378304t^{18} + 25986966113746944t^{16} +
1662158987329536t^{14} + 67844840620032t^{12} + 1824948486144t^{10} +
31619579904t^{8} + 296828928t^{6} + 466560t^{4} - 12960t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 - 8018112x + 8735863860$, with conductor $13872$ |
| Generic density of odd order reductions |
$299/2688$ |