Curve name | $X_{85f}$ | ||||||||||||
Index | $48$ | ||||||||||||
Level | $16$ | ||||||||||||
Genus | $0$ | ||||||||||||
Does the subgroup contain $-I$? | No | ||||||||||||
Generating matrices | $ \left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$ | ||||||||||||
Images in lower levels |
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Meaning/Special name | |||||||||||||
Chosen covering | $X_{85}$ | ||||||||||||
Curves that $X_{85f}$ minimally covers | |||||||||||||
Curves that minimally cover $X_{85f}$ | |||||||||||||
Curves that minimally cover $X_{85f}$ and have infinitely many rational points. | |||||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -27t^{16} + 540t^{14} - 2916t^{12} - 5184t^{10} + 105840t^{8} - 399168t^{6} + 627264t^{4} - 345600t^{2} - 27648\] \[B(t) = -54t^{24} + 1620t^{22} - 28512t^{20} + 348192t^{18} - 2840832t^{16} + 15054336t^{14} - 50754816t^{12} + 103389696t^{10} - 107039232t^{8} + 7326720t^{6} + 91238400t^{4} - 62373888t^{2} + 1769472\] | ||||||||||||
Info about rational points | |||||||||||||
Comments on finding rational points | None | ||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - 960204x - 1015494032$, with conductor $28224$ | ||||||||||||
Generic density of odd order reductions | $635/5376$ |