| Curve name |
$X_{207i}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 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 $X_{207i}$ minimally covers |
|
| Curves that minimally cover $X_{207i}$ |
|
| Curves that minimally cover $X_{207i}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -108t^{24} - 5184t^{23} - 6912t^{22} + 3193344t^{21} + 73972224t^{20} +
778899456t^{19} + 3989716992t^{18} + 2577235968t^{17} - 90228473856t^{16} -
511632211968t^{15} - 777605087232t^{14} + 3516408004608t^{13} +
19299984998400t^{12} + 28131264036864t^{11} - 49766725582848t^{10} -
261955692527616t^{9} - 369575828914176t^{8} + 84450868199424t^{7} +
1045880371150848t^{6} + 1633470551949312t^{5} + 1241047980048384t^{4} +
428603376402432t^{3} - 7421703487488t^{2} - 44530220924928t - 7421703487488\]
\[B(t) = 432t^{36} + 31104t^{35} + 1907712t^{34} + 80372736t^{33} +
2035196928t^{32} + 30985224192t^{31} + 264766095360t^{30} + 605499162624t^{29} -
13660083191808t^{28} - 183139842392064t^{27} - 1069050771800064t^{26} -
2199326520508416t^{25} + 12591141023121408t^{24} + 112070128916496384t^{23} +
315454215872839680t^{22} - 292799364508680192t^{21} - 5051413852437086208t^{20}
- 14527984924650110976t^{19} + 116223879397200887808t^{17} +
323290486555973517312t^{16} + 149913274628444258304t^{15} -
1292100468215151329280t^{14} - 3672313984335753510912t^{13} -
3300692072365138378752t^{12} + 4612322011137265631232t^{11} +
17935695713456382541824t^{10} + 24580613552140915310592t^{9} +
14667402642363663777792t^{8} - 5201198202450931089408t^{7} -
18194587530573077544960t^{6} - 17034307144175107178496t^{5} -
8950890748599978688512t^{4} - 2827864249221462884352t^{3} -
536973190770638979072t^{2} - 70039981404865953792t - 7782220156096217088\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 3982655x + 138589366943$, with conductor $47040$ |
| Generic density of odd order reductions |
$271/2688$ |