| Curve name |
$X_{226b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 3 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 11 & 11 \\ 12 & 3 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{226}$ |
| Curves that $X_{226b}$ minimally covers |
|
| Curves that minimally cover $X_{226b}$ |
|
| Curves that minimally cover $X_{226b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -3564t^{32} - 88128t^{31} - 800064t^{30} - 2229120t^{29} +
14097024t^{28} + 134638848t^{27} + 323847936t^{26} - 988789248t^{25} -
8550358272t^{24} - 18531652608t^{23} + 32398396416t^{22} + 313799436288t^{21} +
855828633600t^{20} + 527502938112t^{19} - 4596406456320t^{18} -
21383917608960t^{17} - 55056817096704t^{16} - 100407001006080t^{15} -
137527648174080t^{14} - 142248065531904t^{13} - 106908432629760t^{12} -
50300615983104t^{11} - 2987281022976t^{10} + 17578568318976t^{9} +
16055072391168t^{8} + 7268445978624t^{7} + 1212605005824t^{6} -
636033171456t^{5} - 533403795456t^{4} - 190678302720t^{3} - 39437991936t^{2} -
4586471424t - 233570304\]
\[B(t) = 81648t^{48} + 3017088t^{47} + 45567360t^{46} + 321898752t^{45} +
326778624t^{44} - 13165161984t^{43} - 112051911168t^{42} - 346729559040t^{41} +
718858022400t^{40} + 11477684938752t^{39} + 49025054656512t^{38} +
76075655196672t^{37} - 278725138698240t^{36} - 2290028887941120t^{35} -
8241350089236480t^{34} - 17222563095035904t^{33} - 3186296647446528t^{32} +
155189034483646464t^{31} + 798926464888012800t^{30} + 2429545667999956992t^{29}
+ 4577511419798224896t^{28} + 2139614524879405056t^{27} -
20891300786264604672t^{26} - 95301949961009627136t^{25} -
257483368125498654720t^{24} - 520343985360089382912t^{23} -
837405411543094394880t^{22} - 1095412709814630875136t^{21} -
1163787006690629517312t^{20} - 981330926754255077376t^{19} -
612062871778077179904t^{18} - 213458880970339909632t^{17} +
64494374282207428608t^{16} + 166414170138882342912t^{15} +
138462146838512271360t^{14} + 66761729722244136960t^{13} +
12513730463753306112t^{12} - 8824262876895117312t^{11} -
9449870858832052224t^{10} - 4482763270058409984t^{9} - 1067959050269884416t^{8}
+ 83592541668114432t^{7} + 187906454106144768t^{6} + 86200658008473600t^{5} +
23929753618612224t^{4} + 4449905556848640t^{3} + 546756316102656t^{2} +
40442485800960t + 1369826131968\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 248569464313004x + 1506985808355141712752$, with conductor
$47524672$ |
| Generic density of odd order reductions |
$13411/86016$ |