| Curve name |
$X_{228f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{228}$ |
| Curves that $X_{228f}$ minimally covers |
|
| Curves that minimally cover $X_{228f}$ |
|
| Curves that minimally cover $X_{228f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 1944t^{24} + 134784t^{23} + 4202496t^{22} + 79045632t^{21} +
1003815936t^{20} + 9035587584t^{19} + 58033373184t^{18} + 251602993152t^{17} +
535696146432t^{16} - 1611125489664t^{15} - 21095843364864t^{14} -
108621912342528t^{13} - 367214645477376t^{12} - 868975298740224t^{11} -
1350133975351296t^{10} - 824896250707968t^{9} + 2194211415785472t^{8} +
8244526879604736t^{7} + 15213100579946496t^{6} + 18949000572960768t^{5} +
16841236782514176t^{4} + 10609325135364096t^{3} + 4512395720392704t^{2} +
1157785744048128t + 133590662774784\]
\[B(t) = -40824t^{36} - 2985984t^{35} - 97635456t^{34} - 1757528064t^{33} -
14342303232t^{32} + 127874433024t^{31} + 6166845259776t^{30} +
110870429958144t^{29} + 1352493952008192t^{28} + 12560963400105984t^{27} +
92759349685911552t^{26} + 554683331417997312t^{25} + 2694984873541632000t^{24} +
10517681266659164160t^{23} + 31776679396468850688t^{22} +
66603386247359496192t^{21} + 52945201293019840512t^{20} -
241599423328592855040t^{19} - 1152356389556116783104t^{18} -
1932795386628742840320t^{17} + 3388492882753269792768t^{16} +
34100933758648062050304t^{15} + 130157278807936412418048t^{14} +
344643379745887491194880t^{13} + 706474114689697579008000t^{12} +
1163255257849915898855424t^{11} + 1556243645700070264799232t^{10} +
1685903969053380131684352t^{9} + 1452229322978244541022208t^{8} +
952369741527374257717248t^{7} + 423782379363688708571136t^{6} +
70299713002575665037312t^{5} - 63078116690693223088128t^{4} -
61837521360340540981248t^{3} - 27481937703734278619136t^{2} -
6723838214867131564032t - 735419804751092514816\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 24x - 56$, with conductor $576$ |
| Generic density of odd order reductions |
$109/896$ |