| Curve name |
$X_{197e}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{197}$ |
| Curves that $X_{197e}$ minimally covers |
|
| Curves that minimally cover $X_{197e}$ |
|
| Curves that minimally cover $X_{197e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 5076t^{16} - 203904t^{15} - 1095552t^{14} + 2336256t^{13} -
8232192t^{12} - 55019520t^{11} + 45398016t^{10} - 270508032t^{9} +
370538496t^{8} + 1082032128t^{7} + 726368256t^{6} + 3521249280t^{5} -
2107441152t^{4} - 2392326144t^{3} - 4487380992t^{2} + 3340763136t + 332660736\]
\[B(t) = 1019088t^{24} + 8066304t^{23} - 40414464t^{22} + 291271680t^{21} +
4044225024t^{20} + 875556864t^{19} - 4345270272t^{18} + 82544541696t^{17} -
241048866816t^{16} + 126959616000t^{15} + 1119470616576t^{14} -
823144218624t^{13} + 14415230140416t^{12} + 3292576874496t^{11} +
17911529865216t^{10} - 8125415424000t^{9} - 61708509904896t^{8} -
84525610696704t^{7} - 17798227034112t^{6} - 14345123659776t^{5} +
265042331172864t^{4} - 76355123281920t^{3} - 42377637003264t^{2} -
33832531132416t + 17097459499008\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 + 63x + 1377$, with conductor $192$ |
| Generic density of odd order reductions |
$109/896$ |