Curve name | $X_{226d}$ | ||||||||||||
Index | $96$ | ||||||||||||
Level | $16$ | ||||||||||||
Genus | $0$ | ||||||||||||
Does the subgroup contain $-I$? | No | ||||||||||||
Generating matrices | $ \left[ \begin{matrix} 11 & 11 \\ 12 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 15 & 0 \\ 0 & 1 \end{matrix}\right]$ | ||||||||||||
Images in lower levels |
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Meaning/Special name | |||||||||||||
Chosen covering | $X_{226}$ | ||||||||||||
Curves that $X_{226d}$ minimally covers | |||||||||||||
Curves that minimally cover $X_{226d}$ | |||||||||||||
Curves that minimally cover $X_{226d}$ and have infinitely many rational points. | |||||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -891t^{16} - 7776t^{15} - 18576t^{14} + 38016t^{13} + 267408t^{12} + 107136t^{11} - 3378240t^{10} - 16561152t^{9} - 45469728t^{8} - 85584384t^{7} - 116156160t^{6} - 114849792t^{5} - 81983232t^{4} - 41084928t^{3} - 13630464t^{2} - 2654208t - 228096\] \[B(t) = 10206t^{24} + 132192t^{23} + 563760t^{22} - 470016t^{21} - 15939504t^{20} - 86873472t^{19} - 291598272t^{18} - 703157760t^{17} - 998518752t^{16} + 1481518080t^{15} + 16926340608t^{14} + 69110931456t^{13} + 190561641984t^{12} + 394978332672t^{11} + 640645687296t^{10} + 828529164288t^{9} + 860825387520t^{8} + 718074077184t^{7} + 477110366208t^{6} + 248593121280t^{5} + 99072626688t^{4} + 29057384448t^{3} + 5888360448t^{2} + 732561408t + 41803776\] | ||||||||||||
Info about rational points | |||||||||||||
Comments on finding rational points | None | ||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - 6434507x + 6281938894$, with conductor $84320$ | ||||||||||||
Generic density of odd order reductions | $9827/86016$ |