| Curve name |
$X_{207b}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 9 & 27 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 24 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{207}$ |
| Curves that $X_{207b}$ minimally covers |
|
| Curves that minimally cover $X_{207b}$ |
|
| Curves that minimally cover $X_{207b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{28} - 1296t^{27} - 2592t^{26} + 762048t^{25} + 18645120t^{24} +
218930688t^{23} + 1446128640t^{22} + 4473999360t^{21} - 8445136896t^{20} -
150641344512t^{19} - 654791344128t^{18} - 907540955136t^{17} +
4010303029248t^{16} + 24341253193728t^{15} + 51034403635200t^{14} -
7484215394304t^{13} - 317498427703296t^{12} - 814256542973952t^{11} -
761478374227968t^{10} + 846219152719872t^{9} + 3654899057295360t^{8} +
5345713865097216t^{7} + 4203003456258048t^{6} + 1608654230913024t^{5} +
35253091565568t^{4} - 163277476724736t^{3} - 29686813949952t^{2}\]
\[B(t) = -54t^{42} - 3888t^{41} - 241056t^{40} - 10217664t^{39} -
264871296t^{38} - 4295960064t^{37} - 42977765376t^{36} - 210655199232t^{35} +
735926943744t^{34} + 22524538060800t^{33} + 200608395558912t^{32} +
985373276110848t^{31} + 1978817014923264t^{30} - 8868427305320448t^{29} -
88665439982321664t^{28} - 321310758015074304t^{27} - 279380339346898944t^{26} +
2805499175544815616t^{25} + 14689451244804636672t^{24} +
28818268225787658240t^{23} - 25235310639499444224t^{22} -
311922964830892326912t^{21} - 827318285604879335424t^{20} -
587831300812217253888t^{19} + 2843300115942438076416t^{18} +
10685903310986485432320t^{17} + 15984104276431000829952t^{16} +
554279653325414596608t^{15} - 49416991593669781880832t^{14} -
113736009507646627381248t^{13} - 135421954454387247022080t^{12} -
74294499884695516348416t^{11} + 35217023749080389517312t^{10} +
109638592697262164410368t^{9} + 106614713777859028058112t^{8} +
59538361692973864845312t^{7} + 20042135234506295083008t^{6} +
3957258949374926389248t^{5} + 513626530302350327808t^{4} +
62257761248769736704t^{3}\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 4571925x - 170457227750$, with conductor $25200$ |
| Generic density of odd order reductions |
$139/1344$ |