| Curve name |
$X_{207g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207g}$ minimally covers |
|
| Curves that minimally cover $X_{207g}$ |
|
| Curves that minimally cover $X_{207g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{22} - 1188t^{21} + 2052t^{20} + 747360t^{19} + 15584832t^{18} +
159584256t^{17} + 918438912t^{16} + 2520170496t^{15} - 3037519872t^{14} -
55277420544t^{13} - 223667453952t^{12} - 442219364352t^{11} - 194401271808t^{10}
+ 1290327293952t^{9} + 3761925783552t^{8} + 5229256900608t^{7} +
4085470199808t^{6} + 1567327518720t^{5} + 34426847232t^{4} - 159450660864t^{3} -
28991029248t^{2}\]
\[B(t) = 54t^{33} + 3564t^{32} + 219672t^{31} + 8905680t^{30} + 211854528t^{29}
+ 3050618112t^{28} + 25744103424t^{27} + 82835675136t^{26} - 823372480512t^{25}
- 13766454558720t^{24} - 101678002569216t^{23} - 457776036642816t^{22} -
1169279731630080t^{21} - 249857288699904t^{20} + 11198407108460544t^{19} +
47678885424267264t^{18} + 89587256867684352t^{17} - 15990866476793856t^{16} -
598671222594600960t^{15} - 1875050646088974336t^{14} - 3331784788188069888t^{13}
- 3608793463841095680t^{12} - 1726737244250701824t^{11} +
1389752014262501376t^{10} + 3455315070966300672t^{9} + 3275576255906316288t^{8}
+ 1819816538539032576t^{7} + 611993669578260480t^{6} + 120765959148404736t^{5} +
15674637765574656t^{4} + 1899956092796928t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 11430x + 21304296$, with conductor $630$ |
| Generic density of odd order reductions |
$271/2688$ |