| Curve name |
$X_{197h}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{197}$ |
| Curves that $X_{197h}$ minimally covers |
|
| Curves that minimally cover $X_{197h}$ |
|
| Curves that minimally cover $X_{197h}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 1269t^{16} - 50976t^{15} - 273888t^{14} + 584064t^{13} - 2058048t^{12}
- 13754880t^{11} + 11349504t^{10} - 67627008t^{9} + 92634624t^{8} +
270508032t^{7} + 181592064t^{6} + 880312320t^{5} - 526860288t^{4} -
598081536t^{3} - 1121845248t^{2} + 835190784t + 83165184\]
\[B(t) = 127386t^{24} + 1008288t^{23} - 5051808t^{22} + 36408960t^{21} +
505528128t^{20} + 109444608t^{19} - 543158784t^{18} + 10318067712t^{17} -
30131108352t^{16} + 15869952000t^{15} + 139933827072t^{14} - 102893027328t^{13}
+ 1801903767552t^{12} + 411572109312t^{11} + 2238941233152t^{10} -
1015676928000t^{9} - 7713563738112t^{8} - 10565701337088t^{7} -
2224778379264t^{6} - 1793140457472t^{5} + 33130291396608t^{4} -
9544390410240t^{3} - 5297204625408t^{2} - 4229066391552t + 2137182437376\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 16x + 180$, with conductor $48$ |
| Generic density of odd order reductions |
$53/896$ |