Curve name | $X_{45}$ | |||||||||
Index | $12$ | |||||||||
Level | $8$ | |||||||||
Genus | $0$ | |||||||||
Does the subgroup contain $-I$? | Yes | |||||||||
Generating matrices | $ \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 5 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 5 \\ 0 & 1 \end{matrix}\right]$ | |||||||||
Images in lower levels |
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Meaning/Special name | ||||||||||
Chosen covering | $X_{11}$ | |||||||||
Curves that $X_{45}$ minimally covers | $X_{11}$ | |||||||||
Curves that minimally cover $X_{45}$ | $X_{61}$, $X_{69}$, $X_{70}$, $X_{73}$, $X_{77}$, $X_{81}$, $X_{94}$, $X_{97}$, $X_{109}$, $X_{110}$, $X_{111}$, $X_{112}$, $X_{130}$, $X_{136}$, $X_{140}$, $X_{143}$, $X_{149}$, $X_{150}$, $X_{151}$, $X_{153}$ | |||||||||
Curves that minimally cover $X_{45}$ and have infinitely many rational points. | $X_{61}$, $X_{69}$, $X_{70}$, $X_{73}$, $X_{77}$, $X_{81}$, $X_{94}$, $X_{97}$, $X_{109}$, $X_{110}$, $X_{111}$, $X_{112}$, $X_{150}$, $X_{153}$ | |||||||||
Model | \[\mathbb{P}^{1}, \mathbb{Q}(X_{45}) = \mathbb{Q}(f_{45}), f_{11} = -f_{45}^{2} + 8\] | |||||||||
Info about rational points | None | |||||||||
Comments on finding rational points | None | |||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 + xy = x^3 - x^2 + 258x + 1791$, with conductor $1575$ | |||||||||
Generic density of odd order reductions | $2659/10752$ |