Curve name | $X_{302}$ | ||||||||||||
Index | $48$ | ||||||||||||
Level | $16$ | ||||||||||||
Genus | $1$ | ||||||||||||
Does the subgroup contain $-I$? | Yes | ||||||||||||
Generating matrices | $ \left[ \begin{matrix} 13 & 7 \\ 2 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 13 & 10 \\ 2 & 3 \end{matrix}\right]$ | ||||||||||||
Images in lower levels |
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Meaning/Special name | |||||||||||||
Chosen covering | $X_{63}$ | ||||||||||||
Curves that $X_{302}$ minimally covers | $X_{63}$, $X_{105}$, $X_{166}$ | ||||||||||||
Curves that minimally cover $X_{302}$ | |||||||||||||
Curves that minimally cover $X_{302}$ and have infinitely many rational points. | |||||||||||||
Model | \[y^2 = x^3 + x^2 - 13x - 21\] | ||||||||||||
Info about rational points | $X_{302}(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}$ | ||||||||||||
Comments on finding rational points | None | ||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 + x^2 - 7859386910x - 268185327947856$, with conductor $30631008$ | ||||||||||||
Generic density of odd order reductions | $42979/172032$ |