| Curve name |
$X_{189e}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{189}$ |
| Curves that $X_{189e}$ minimally covers |
|
| Curves that minimally cover $X_{189e}$ |
|
| Curves that minimally cover $X_{189e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 864t^{15} - 12096t^{14} - 96768t^{13} - 580608t^{12} -
3870720t^{11} - 27095040t^{10} - 142884864t^{9} - 500539392t^{8} -
1143078912t^{7} - 1734082560t^{6} - 1981808640t^{5} - 2378170368t^{4} -
3170893824t^{3} - 3170893824t^{2} - 1811939328t - 452984832\]
\[B(t) = -54t^{24} - 2592t^{23} - 57024t^{22} - 760320t^{21} - 6386688t^{20} -
25546752t^{19} + 127733760t^{18} + 2934226944t^{17} + 24999321600t^{16} +
138195763200t^{15} + 563838713856t^{14} + 1862107398144t^{13} +
5461864611840t^{12} + 14896859185152t^{11} + 36085677686784t^{10} +
70756230758400t^{9} + 102397221273600t^{8} + 96148748500992t^{7} +
33484638781440t^{6} - 53575422050304t^{5} - 107150844100608t^{4} -
102048422952960t^{3} - 61229053771776t^{2} - 22265110462464t - 3710851743744\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 120800x + 15840000$, with conductor $1680$ |
| Generic density of odd order reductions |
$19/336$ |