Curve name | $X_{2b}$ | |||||||||
Index | $4$ | |||||||||
Level | $8$ | |||||||||
Genus | $0$ | |||||||||
Does the subgroup contain $-I$? | No | |||||||||
Generating matrices | $ \left[ \begin{matrix} 1 & 1 \\ 3 & 0 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right]$ | |||||||||
Images in lower levels |
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Meaning/Special name | ||||||||||
Chosen covering | $X_{2}$ | |||||||||
Curves that $X_{2b}$ minimally covers | ||||||||||
Curves that minimally cover $X_{2b}$ | ||||||||||
Curves that minimally cover $X_{2b}$ and have infinitely many rational points. | ||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -108t^{6} - 559872t^{4} - 967458816t^{2} - 557256278016\] \[B(t) = 432t^{9} + 2985984t^{7} + 7739670528t^{5} + 8916100448256t^{3} + 3851755393646592t\] | |||||||||
Info about rational points | ||||||||||
Comments on finding rational points | None | |||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 + xy + y = x^3 - 4345x + 64840$, with conductor $1922$ | |||||||||
Generic density of odd order reductions | $7681/10752$ |