Curve name | $X_{219g}$ | ||||||||||||
Index | $96$ | ||||||||||||
Level | $16$ | ||||||||||||
Genus | $0$ | ||||||||||||
Does the subgroup contain $-I$? | No | ||||||||||||
Generating matrices | $ \left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right]$ | ||||||||||||
Images in lower levels |
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Meaning/Special name | |||||||||||||
Chosen covering | $X_{219}$ | ||||||||||||
Curves that $X_{219g}$ minimally covers | |||||||||||||
Curves that minimally cover $X_{219g}$ | |||||||||||||
Curves that minimally cover $X_{219g}$ and have infinitely many rational points. | |||||||||||||
Model | $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by \[y^2 = x^3 + A(t)x + B(t), \text{ where}\] \[A(t) = -5211t^{16} - 13392t^{15} - 16632t^{14} - 60048t^{13} - 102708t^{12} - 127440t^{11} - 289224t^{10} - 80784t^{9} - 506466t^{8} + 80784t^{7} - 289224t^{6} + 127440t^{5} - 102708t^{4} + 60048t^{3} - 16632t^{2} + 13392t - 5211\] \[B(t) = -144774t^{24} - 558576t^{23} - 1045224t^{22} - 3299184t^{21} - 7268940t^{20} - 14034384t^{19} - 23074632t^{18} - 42888528t^{17} - 57334554t^{16} - 63764064t^{15} - 127667664t^{14} - 32169312t^{13} - 159922728t^{12} + 32169312t^{11} - 127667664t^{10} + 63764064t^{9} - 57334554t^{8} + 42888528t^{7} - 23074632t^{6} + 14034384t^{5} - 7268940t^{4} + 3299184t^{3} - 1045224t^{2} + 558576t - 144774\] | ||||||||||||
Info about rational points | |||||||||||||
Comments on finding rational points | None | ||||||||||||
Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 + x^2 - 64x - 220$, with conductor $48$ | ||||||||||||
Generic density of odd order reductions | $53/896$ |