| Curve name |
$X_{123b}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 15 & 15 \\ 6 & 3 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{123}$ |
| Curves that $X_{123b}$ minimally covers |
|
| Curves that minimally cover $X_{123b}$ |
|
| Curves that minimally cover $X_{123b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -189t^{16} - 4320t^{15} - 41904t^{14} - 235008t^{13} - 892944t^{12} -
2547072t^{11} - 5711040t^{10} - 9856512t^{9} - 12141792t^{8} - 9220608t^{7} -
2315520t^{6} + 2764800t^{5} + 2965248t^{4} + 940032t^{3} - 193536t^{2} - 221184t
- 48384\]
\[B(t) = 918t^{24} + 29808t^{23} + 417312t^{22} + 3256416t^{21} + 14760144t^{20}
+ 30165696t^{19} - 71798400t^{18} - 814271616t^{17} - 3389120352t^{16} -
9109559808t^{15} - 17488825344t^{14} - 24583357440t^{13} - 24853989888t^{12} -
16374638592t^{11} - 3728166912t^{10} + 5567311872t^{9} + 7500335616t^{8} +
4125634560t^{7} + 321822720t^{6} - 1083580416t^{5} - 680472576t^{4} -
100417536t^{3} + 61046784t^{2} + 29196288t + 3760128\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 76162x - 8064798$, with conductor $1664$ |
| Generic density of odd order reductions |
$45667/172032$ |