| Curve name |
$X_{207e}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 0 \\ 24 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 9 & 27 \\ 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 $X_{207e}$ minimally covers |
|
| Curves that minimally cover $X_{207e}$ |
|
| Curves that minimally cover $X_{207e}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{26} - 1188t^{25} + 2052t^{24} + 749088t^{23} + 15657408t^{22} +
159314688t^{21} + 871395840t^{20} + 1614034944t^{19} - 11619237888t^{18} -
99396550656t^{17} - 310057058304t^{16} - 58470432768t^{15} + 3189650227200t^{14}
+ 12170626596864t^{13} + 16041212116992t^{12} - 29041310564352t^{11} -
167626584096768t^{10} - 308513473560576t^{9} - 138365122904064t^{8} +
586416044113920t^{7} + 1516375784816640t^{6} + 1836523720802304t^{5} +
1251252822343680t^{4} + 409121404747776t^{3} - 11132555231232t^{2} -
44530220924928t - 7421703487488\]
\[B(t) = 54t^{39} + 3564t^{38} + 219672t^{37} + 8900496t^{36} + 211522752t^{35}
+ 3030172416t^{34} + 24928929792t^{33} + 64059393024t^{32} - 1081127264256t^{31}
- 15765190410240t^{30} - 105924913201152t^{29} - 368185278726144t^{28} +
33037641842688t^{27} + 7621814859595776t^{26} + 40763486141153280t^{25} +
86770633421094912t^{24} - 143513501174857728t^{23} - 1615856026572029952t^{22} -
4873796325482692608t^{21} - 3154413829937430528t^{20} +
28818268225787658240t^{19} + 117515609958437093376t^{18} +
179551947234868199424t^{17} - 143042733745612259328t^{16} -
1316088864829744349184t^{15} - 2905389137340716285952t^{14} -
2324805007525923520512t^{13} + 4149880060480352944128t^{12} +
16531820293939336839168t^{11} + 26925203069642458791936t^{10} +
24185538582160814899200t^{9} + 6321564311251423592448t^{8} -
14476115062940869066752t^{7} - 23627276383370386341888t^{6} -
18893832171317316550656t^{5} - 9319330234115158966272t^{4} -
2876016736437308227584t^{3} - 542809855887711141888t^{2} - 70039981404865953792t
- 7782220156096217088\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + 1555724x + 33835100198$, with conductor $7350$ |
| Generic density of odd order reductions |
$1091/10752$ |