| Curve name |
$X_{123d}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 13 & 13 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{123}$ |
| Curves that $X_{123d}$ minimally covers |
|
| Curves that minimally cover $X_{123d}$ |
|
| Curves that minimally cover $X_{123d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -756t^{16} - 17280t^{15} - 167616t^{14} - 940032t^{13} - 3571776t^{12}
- 10188288t^{11} - 22844160t^{10} - 39426048t^{9} - 48567168t^{8} -
36882432t^{7} - 9262080t^{6} + 11059200t^{5} + 11860992t^{4} + 3760128t^{3} -
774144t^{2} - 884736t - 193536\]
\[B(t) = -7344t^{24} - 238464t^{23} - 3338496t^{22} - 26051328t^{21} -
118081152t^{20} - 241325568t^{19} + 574387200t^{18} + 6514172928t^{17} +
27112962816t^{16} + 72876478464t^{15} + 139910602752t^{14} + 196666859520t^{13}
+ 198831919104t^{12} + 130997108736t^{11} + 29825335296t^{10} - 44538494976t^{9}
- 60002684928t^{8} - 33005076480t^{7} - 2574581760t^{6} + 8668643328t^{5} +
5443780608t^{4} + 803340288t^{3} - 488374272t^{2} - 233570304t - 30081024\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 304649x + 64823033$, with conductor $1664$ |
| Generic density of odd order reductions |
$12833/57344$ |