| Curve name |
$X_{40d}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 14 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{40}$ |
| Curves that $X_{40d}$ minimally covers |
|
| Curves that minimally cover $X_{40d}$ |
|
| Curves that minimally cover $X_{40d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -70778880t^{12} - 169869312t^{11} - 130940928t^{10} - 21233664t^{9} +
15261696t^{8} + 5308416t^{7} + 552960t^{6} + 663552t^{5} + 238464t^{4} -
41472t^{3} - 31968t^{2} - 5184t - 270\]
\[B(t) = 202937204736t^{18} + 652298158080t^{17} + 641426522112t^{16} -
97844723712t^{15} - 625119068160t^{14} - 411763212288t^{13} - 15174991872t^{12}
+ 102431195136t^{11} + 45312638976t^{10} - 5664079872t^{8} - 1600487424t^{7} +
29638656t^{6} + 100528128t^{5} + 19077120t^{4} + 373248t^{3} - 305856t^{2} -
38880t - 1512\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 1731x - 20257$, with conductor $12544$ |
| Generic density of odd order reductions |
$106595/344064$ |