| Curve name |
$X_{219f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{219}$ |
| Curves that $X_{219f}$ minimally covers |
|
| Curves that minimally cover $X_{219f}$ |
|
| Curves that minimally cover $X_{219f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -46899t^{24} + 4536t^{23} - 99252t^{22} - 712584t^{21} - 141318t^{20} -
1824984t^{19} - 4402404t^{18} - 489240t^{17} - 19534365t^{16} + 6711984t^{15} -
45292392t^{14} + 6093360t^{13} - 53065044t^{12} - 6093360t^{11} - 45292392t^{10}
- 6711984t^{9} - 19534365t^{8} + 489240t^{7} - 4402404t^{6} + 1824984t^{5} -
141318t^{4} + 712584t^{3} - 99252t^{2} - 4536t - 46899\]
\[B(t) = -3908898t^{36} + 554040t^{35} - 12195684t^{34} - 90945720t^{33} -
1694034t^{32} - 420634944t^{31} - 497906784t^{30} - 1595422656t^{29} -
1731744360t^{28} - 6348776544t^{27} - 9067830768t^{26} - 6801918624t^{25} -
48120356232t^{24} + 12379563072t^{23} - 126428068512t^{22} + 28502988480t^{21} -
206315835804t^{20} + 18027874512t^{19} - 240335075160t^{18} - 18027874512t^{17}
- 206315835804t^{16} - 28502988480t^{15} - 126428068512t^{14} -
12379563072t^{13} - 48120356232t^{12} + 6801918624t^{11} - 9067830768t^{10} +
6348776544t^{9} - 1731744360t^{8} + 1595422656t^{7} - 497906784t^{6} +
420634944t^{5} - 1694034t^{4} + 90945720t^{3} - 12195684t^{2} - 554040t -
3908898\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 579x - 5362$, with conductor $72$ |
| Generic density of odd order reductions |
$299/2688$ |