| Curve name |
$X_{213a}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 0 \\ 24 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{213}$ |
| Curves that $X_{213a}$ minimally covers |
|
| Curves that minimally cover $X_{213a}$ |
|
| Curves that minimally cover $X_{213a}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{30} - 594t^{28} - 5049t^{26} - 20628t^{24} - 40419t^{22} -
28782t^{20} + 17847t^{18} + 44712t^{16} + 17847t^{14} - 28782t^{12} -
40419t^{10} - 20628t^{8} - 5049t^{6} - 594t^{4} - 27t^{2}\]
\[B(t) = 54t^{45} + 1782t^{43} + 24948t^{41} + 192564t^{39} + 890190t^{37} +
2492046t^{35} + 3956688t^{33} + 2432592t^{31} - 2664900t^{29} - 6870852t^{27} -
7533000t^{25} - 7533000t^{23} - 6870852t^{21} - 2664900t^{19} + 2432592t^{17} +
3956688t^{15} + 2492046t^{13} + 890190t^{11} + 192564t^{9} + 24948t^{7} +
1782t^{5} + 54t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 7378388105x + 243944772676647$, with conductor
$130050$ |
| Generic density of odd order reductions |
$299/2688$ |