| Curve name |
$X_{204d}$ |
| Index |
$96$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{204}$ |
| Curves that $X_{204d}$ minimally covers |
|
| Curves that minimally cover $X_{204d}$ |
|
| Curves that minimally cover $X_{204d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -17739t^{24} - 320112t^{23} - 3144528t^{22} - 18767808t^{21} -
84271968t^{20} - 275035392t^{19} - 733798656t^{18} - 1430369280t^{17} -
2579662080t^{16} - 3808124928t^{15} - 7866851328t^{14} - 5472092160t^{13} -
8468250624t^{12} + 21888368640t^{11} - 125869621248t^{10} + 243719995392t^{9} -
660393492480t^{8} + 1464698142720t^{7} - 3005639294976t^{6} + 4506179862528t^{5}
- 5522847694848t^{4} + 4919868260352t^{3} - 3297276592128t^{2} + 1342647042048t
- 297611034624\]
\[B(t) = -867510t^{36} - 25345872t^{35} - 349986096t^{34} - 3283257024t^{33} -
22604196960t^{32} - 121327994880t^{31} - 519456098304t^{30} -
1836846637056t^{29} - 5448260745216t^{28} - 14046538088448t^{27} -
31155737149440t^{26} - 61732891459584t^{25} - 103622008209408t^{24} -
177813547646976t^{23} - 273955331506176t^{22} - 509882161692672t^{21} -
580576552943616t^{20} - 510617469321216t^{19} - 787850868031488t^{18} +
2042469877284864t^{17} - 9289224847097856t^{16} + 32632458348331008t^{15} -
70132564865581056t^{14} + 182081072790503424t^{13} - 424435745625735168t^{12} +
1011431693673824256t^{11} - 2041822389825699840t^{10} + 3682215680658112512t^{9}
- 5712915459175612416t^{8} + 7704293197190529024t^{7} - 8715027163763441664t^{6}
+ 8142183907794616320t^{5} - 6067767918471413760t^{4} + 3525370385610571776t^{3}
- 1503178836374716416t^{2} + 435438765314408448t - 59614833263247360\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 219x - 1190$, with conductor $144$ |
| Generic density of odd order reductions |
$299/2688$ |