| Curve name |
$X_{82b}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 10 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 14 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 14 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{82}$ |
| Curves that $X_{82b}$ minimally covers |
|
| Curves that minimally cover $X_{82b}$ |
|
| Curves that minimally cover $X_{82b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -17010t^{20} - 312336t^{19} - 2677320t^{18} - 14580000t^{17} -
57887784t^{16} - 181336320t^{15} - 470240640t^{14} - 1039993344t^{13} -
1999702080t^{12} - 3384737280t^{11} - 5078502144t^{10} - 6769474560t^{9} -
7998808320t^{8} - 8319946752t^{7} - 7523850240t^{6} - 5802762240t^{5} -
3704818176t^{4} - 1866240000t^{3} - 685393920t^{2} - 159916032t - 17418240\]
\[B(t) = 851472t^{30} + 23397984t^{29} + 306708768t^{28} + 2575694592t^{27} +
15696244800t^{26} + 74432151936t^{25} + 286950169728t^{24} + 925319614464t^{23}
+ 2541686936832t^{22} + 6010028969472t^{21} + 12279188224512t^{20} +
21583379460096t^{19} + 32068140143616t^{18} + 38352876779520t^{17} +
31244579629056t^{16} - 62489159258112t^{14} - 153411507118080t^{13} -
256545121148928t^{12} - 345334071361536t^{11} - 392934023184384t^{10} -
384641854046208t^{9} - 325335927914496t^{8} - 236881821302784t^{7} -
146918486900736t^{6} - 76218523582464t^{5} - 32145909350400t^{4} -
10550045048832t^{3} - 2512558227456t^{2} - 383352569856t - 27901034496\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 210x + 1168$, with conductor $2304$ |
| Generic density of odd order reductions |
$403/1792$ |