短型魏氏椭圆曲线的复乘自同态

青青子衿

\begin{align*}
\boxed{\qquad\begin{matrix}
y^2=x^3-30x+56\\
\\
\varphi_{\tiny(i\sqrt{2},-35,98)}\colon\left\{
\begin{split}
&\begin{split}
X&=\tfrac{1}{(\sqrt{2}\,i)^2}\left(x+\tfrac{18}{x-4}\right)\\
Y&=\tfrac{y}{(\sqrt{2}\,i)^3}\left(1-\tfrac{18}{(x-4)^2}\right)
\end{split}\\
\end{split}
\right.\\
\\
\begin{split}
(\varphi_{\tiny(i\sqrt{2},-30,56)}\circ\varphi_{\tiny(i\sqrt{2},-30,56)}\circ\mathcal{P})\oplus(2\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi_{\tiny(i\sqrt{2},-30,56)}^2+2)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-30 X+56\\
\end{matrix}\qquad}
\end{align*}

\begin{align*}
\boxed{\qquad\begin{matrix}
y^2=x^3-35x+98\\
\\
\varphi_{\tiny(\frac{1+i \sqrt{7}}{2},-35,98)}\colon\left\{
\begin{split}
&\begin{split}
X&=\tfrac{1}{\big(\frac{1+i \sqrt{7}}{2}\big)^2}\left(x+\tfrac{-\>\!\frac{7+21i\sqrt{7}}{2}}{x-\frac{7-i \sqrt{7}}{2}}\right)\\
Y&=\tfrac{y}{\big(\frac{1+i \sqrt{7}}{2}\big)^3}\left(1-\tfrac{-\>\!\frac{7+21i\sqrt{7}}{2}}{(x-\frac{7-i \sqrt{7}}{2})^2}\right)
\end{split}\\
\end{split}
\right.\\
\\
\begin{split}
(\varphi_{\tiny(\frac{1+i \sqrt{7}}{2},-35,98)}\circ\varphi_{\tiny(\frac{1+i \sqrt{7}}{2},-35,98)}\circ\mathcal{P})\ominus(\varphi_{\tiny(\frac{1+i \sqrt{7}}{2},-35,98)}\circ\mathcal{P})\oplus(2\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi_{\tiny(\frac{1+i \sqrt{7}}{2},-35,98)}^2-\varphi_{\tiny(\frac{1+i \sqrt{7}}{2},-35,98)}+2)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-35 X+98
\end{matrix}\qquad}
\end{align*}

\begin{align*}
\boxed{\qquad\begin{matrix}
y^2=x^3-35x+98\\
\\
\varphi_{\tiny(i\sqrt{7},-35,98)}\colon\left\{
\begin{split}
&\begin{split}
X&=\tfrac{1}{(\sqrt{7}\,i)^2}\left(x+\tfrac{
\phi(x,-35,98)
}{(\psi(x,-35,98))^2}\right)\\
Y&=\tfrac{y}{(\sqrt{7}\,i)^3}\left(1-\tfrac{
\omega(x,-35,98)
}{(\psi(x,-35,98))^3}\right)\\
\end{split}\\

&\scriptsize{
\begin{split}
\psi(x,-35,98)&=x^3-7 x^2-21x+91\\
\phi(x,-35,98)&=112(3x^5+x^4-210x^3\\
&\qquad+1106x^2-2625 x+3773)\\

\omega(x,-35,98)&=112 (3x^7+23x^6-441x^5\\
&\qquad+4571x^4-33383 x^3\\
&\qquad\quad+135093x^2-251811x\\
&\qquad\qquad+80409)
\end{split}}
\end{split}
\right.\\
\\
\begin{split}
(\varphi_{\tiny(i\sqrt{7},-35,98)}\circ\varphi_{\tiny(i\sqrt{7},-35,98)}\circ\mathcal{P})\oplus(7\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi_{\tiny(i\sqrt{7},-35,98)}^2+7)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-35 X+98
\end{matrix}\qquad}
\end{align*}

\begin{align*}
\boxed{\qquad\begin{matrix}
y^2=x^3-264x+1694\\
\\
\varphi_{\tiny(\frac{1+i \sqrt{11}}{2},-264,1694)}\colon\left\{
\begin{split}
&\begin{split}
X&=\tfrac{1}{(\frac{1+i\sqrt{11}}{2})^{2}}\left(x+\tfrac{132(1-i\sqrt{11})x-88(11-14i\sqrt{11})}{(x-(11-i\sqrt{11}))^{2}}\right)\\
Y&=\tfrac{y}{(\frac{1+i\sqrt{11}}{2})^{3}}\left(1-\tfrac{132(1-i\sqrt{11})x-176(11-5i\sqrt{11})}{(x-(11-i\sqrt{11}))^{3}}\right)\\
\end{split}\\
\end{split}
\right.\\
\\
\begin{split}
(\varphi_{\tiny(\frac{1+i \sqrt{11}}{2},-264,1694)}\circ\varphi_{\tiny(\frac{1+i \sqrt{11}}{2},-264,1694)}\circ\mathcal{P})\ominus(\varphi_{\tiny(\frac{1+i \sqrt{11}}{2},-264,1694)}\circ\mathcal{P})\oplus(3\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi_{\tiny(\frac{1+i \sqrt{11}}{2},-264,1694)}^2-\varphi_{\tiny(\frac{1+i \sqrt{11}}{2},-264,1694)}+3)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-264 X+1694
\end{matrix}\qquad}
\end{align*}

\begin{align*}
\boxed{\qquad\begin{matrix}
y^2=x^3-264x+1694\\
\\
\varphi_{\tiny(i\sqrt{11},-264,1694)}\colon\left\{
\begin{split}
&\begin{split}
X&=\tfrac{1}{(\sqrt{11}\,i)^2}\left(x+\tfrac{
\phi(x,-264,1694)
}{(\psi(x,-264,1694))^2}\right)\\
Y&=\tfrac{y}{(\sqrt{11}\,i)^3}\left(1-\tfrac{
\omega(x,-264,1694)
}{(\psi(x,-264,1694))^3}\right)\\
\end{split}\\

&\scriptsize{
\begin{split}
\psi(x,-264,1694)&=x^{5}-44x^{4}+220x^{3}+6776x^{2}\\
&\qquad-71632x+166496\\
\phi(x,-264,1694)&=792(8x^{9}-297x^{8}+792x^{7}+121440x^{6}\\
&\qquad-2509056x^{5}+24480720x^{4}-173775360x^{3}\\
&\qquad\quad+1195983360x^{2}-6093381888x\\
&\qquad\qquad+13564359424)\\

\omega(x,-264,1694)&=1584(4x^{13}-121x^{12}-1452x^{11}\\
&\qquad+155276x^{10}-5672480x^{9}+161172000x^{8}\\
&\qquad\quad-3521762112x^{7}+54722116416x^{6}\\
&\qquad\qquad-572886762624x^{5}+3741656771072x^{4}\\
&\qquad\quad-12010526257664x^{3}-9581302781952x^{2}\\
&\qquad+202938319108096x-464380338847744)
\end{split}}
\end{split}
\right.\\
\\
\begin{split}
(\varphi_{\tiny(i\sqrt{11},-264,1694)}\circ\varphi_{\tiny(i\sqrt{11},-264,1694)}\circ\mathcal{P})\oplus(11\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi_{\tiny(i\sqrt{11},-264,1694)}^2+11)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-264 X+1694
\end{matrix}\qquad}
\end{align*}

青青子衿

\begin{align*}
\boxed{\qquad\begin{matrix}
y^2=x^3-3440 x+77658\\
\\
\varphi_{\tiny(\frac{1+i\sqrt{43}}{2},-3440,77658)}\colon\left\{
\begin{split}
&\begin{split}
X&=\tfrac{1}{(\frac{1+i\sqrt{43}}{2})^2}\left(x+\tfrac{
\phi(x,-3440,77658)
}{(\psi(x,-3440,77658))^2}\right)\\
Y&=\tfrac{y}{(\frac{1+i\sqrt{43}}{2})^3}\left(1-\tfrac{
\omega(x,-3440,77658)
}{(\psi(x,-3440,77658))^3}\right)\\
\end{split}\\

&\scriptsize{
\begin{split}
\psi(x,-3440,77658)&=\Big(x^{5}-6(43-i\sqrt{43})x^{4}+86(279-11i\sqrt{43})x^{3}\\
&\qquad-344(3053-160i\sqrt{43})x^{2}\\
&\qquad\quad+14792(1493-95i\sqrt{43})x\\
&\qquad\qquad-\tfrac{29584}{11}(67123-4951i\sqrt{43})\Big)\\

\phi(x,-3440,77658)&=\tfrac{86(21+i\sqrt{43})}{121}\Big(22(147-29i\sqrt{43})x^{9}\\
&\qquad-33(30487-7067i\sqrt{43})x^{8}\\
&\qquad\quad+7568(18077-4968i\sqrt{43})x^{7}\\
&\qquad\qquad-8944(1190197-392131i\sqrt{43})x^{6}\\
&\qquad\qquad\quad+59168(8793251-3525891i\sqrt{43})x^{5}\\
&\qquad\qquad-59168(278202131-138729472i\sqrt{43})x^{4}\\
&\qquad\quad+20353792(16391301-10500091i\sqrt{43})x^{3}\\
&\qquad-40707584(100678093-87263617i\sqrt{43})x^{2}\\
&\qquad\quad+5251278336(5064431-6516378i\sqrt{43})x\\
&\qquad\qquad-875213056(70930951-166383621i\sqrt{43})\>\!\Big)\\

\omega(x,-3440,77658)&=\tfrac{344(16-5i\sqrt{43}\,)}{1331}\Big(11(697+59i\sqrt{43})x^{13}\\
&\qquad-726(4644+365i\sqrt{43})x^{12}\\
&\qquad\quad+8241552(83+6i\sqrt{43})x^{11}\\
&\qquad\qquad-516(164572997+10766883i\sqrt{43})x^{10}\\
&\qquad\qquad\quad+118336(60907552+3518143i\sqrt{43})x^{9}\\
&\qquad\qquad\qquad-710016(622885530+30614191i\sqrt{43})x^{8}\\
&\qquad\qquad\quad+61061376(330937679+13077950i\sqrt{43})x^{7}\\
&\qquad\qquad-30530688(22822292742+659564693i\sqrt{43})x^{6}\\
&\qquad\quad+1312819584(13819806107+240772331i\sqrt{43})x^{5}\\
&\qquad-7001704448(50337704943+266450842i\sqrt{43})x^{4}\\
&\qquad\quad+150536645632(32983567246-238169247i\sqrt{43})x^{3}\\
&\qquad\qquad-3612879495168(13292107024-263551895i\sqrt{43})x^{2}\\
&\qquad\qquad\quad+12946151524352(22064491369-711380145i\sqrt{43})x\\
&\qquad\qquad\qquad-103569212194816(7614390163-336615144i\sqrt{43})\>\!\Big)
\end{split}}
\end{split}
\right.\\
\\
\begin{split}

(\varphi_{\tiny(\frac{1+i\sqrt{43}}{2},-3440,77658)}\circ\varphi_{\tiny(\frac{1+i\sqrt{43}}{2},-3440,77658)}\circ\mathcal{P})\ominus(\varphi_{\tiny(\frac{1+i\sqrt{43}}{2},-3440,77658)}\circ\mathcal{P})\oplus(11\otimes\mathcal{P})&=\mathcal{O}\\
(\varphi_{\tiny(\frac{1+i\sqrt{43}}{2},-3440,77658)}^2-\varphi_{\tiny(\frac{1+i\sqrt{43}}{2},-3440,77658)}+11)\circ\mathcal{P}&=\mathcal{O}\\
\end{split}\\
\\
Y^2=X^3-3440 X+77658
\end{matrix}\qquad}
\end{align*}

青青子衿

青青子衿 发表于 2024-9-14 09:45
y^2=x^3-3440 x+77658

\begin{gather*}
\begin{gathered}
y^2=x^3+Ax+B\\

A =\scriptsize{ -15 (36 + 4 w_1 - 23 w_2 + 3 w_1 w_2)} \\
B =\scriptsize{ 16 (520 + 24 w_1 + 167 w_2 - 79 w_1 w_2)}\\
w_1=\scriptsize{\sqrt{17}}\qquad\qquad
w_2=\Tiny{\sqrt{2+2 \sqrt{17}}}\\
Y^2=X^3+AX+B\\
\end{gathered}\\
\\
\left\{
\begin{split}
X&=\tfrac{1}{(\sqrt{17}\,i)^2}\left(x+\tfrac{
\sum_{k=0}^{15}\phi_{k}x^{k}
}{(x^8+\sum_{k=0}^{7}\psi_{k}x^{k})^2}\right)\\
Y&=\tfrac{y}{i \sqrt{17}}\cdot\tfrac{\partial{X}}{\partial{x}}\\
\end{split}\right.\\
\\
\quad
\left\{
\begin{split}

\phi_0 &=\scriptsize{ -3623878656 (169008142170402154 - 41842898672281562 w_1}\\&\qquad\qquad\qquad\scriptsize{+ 23900394076652835 w_2 - 5530399938160171 w_1 w_2)} \\
\phi_1 &=\scriptsize{ \frac{16307453952}{17} (-108956551333663360 + 24933019881251456 w_1}\\&\qquad\qquad\qquad\scriptsize{- 56465464505977865 w_2 + 14161262653146657 w_1 w_2)} \\
\phi_2 &=\scriptsize{ -\frac{2038431744}{17} (-26521301599439096 + 4014764402036280 w_1}\\&\qquad\qquad\qquad\scriptsize{+ 61851927343336513 w_2 - 14245977815106457 w_1 w_2)} \\
\phi_3 &=\scriptsize{ \frac{226492416}{17} (77590228805229456 - 21302767110926096 w_1}\\&\qquad\qquad\qquad\scriptsize{- 8973665389921273 w_2 + 2952736648861073 w_1 w_2)} \\
\phi_4 &=\scriptsize{ -\frac{509607936}{17} (-1607923238638538 + 313143517136954 w_1}\\&\qquad\qquad\qquad\scriptsize{- 678901788330707 w_2 + 188682450577371 w_1 w_2)} \\
\phi_5 &=\scriptsize{ \frac{254803968}{17} (-15480192441664 - 3010041502400 w_1}\\&\qquad\qquad\qquad\scriptsize{+ 79158299015593 w_2 - 17077105126081 w_1 w_2)} \\
\phi_6 &=\scriptsize{ -\frac{3538944}{17} (-218386065679880 + 36208854478280 w_1}\\&\qquad\qquad\qquad\scriptsize{- 131419831948385 w_2 + 37199291913017 w_1 w_2)} \\
\phi_7 &=\scriptsize{ \frac{286654464}{17} (153541209280 - 48090254400 w_1}\\&\qquad\qquad\qquad\scriptsize{+ 124109335533 w_2 - 26582433701 w_1 w_2)} \\
\phi_8 &=\scriptsize{ -23887872 (-1156444598 - 234364858 w_1}\\&\qquad\qquad\qquad\scriptsize{- 2666177101 w_2 + 815600133 w_1 w_2)} \\
\phi_9 &=\scriptsize{ \frac{442368}{17} (-78735910720 - 2379819008 w_1}\\&\qquad\qquad\qquad\scriptsize{+ 39269661581 w_2 - 2299200037 w_1 w_2)} \\
\phi_{10} &=\scriptsize{ -497664 (-142290504 + 7630024 w_1}\\&\qquad\qquad\qquad\scriptsize{+ 20679275 w_2 + 4815469 w_1 w_2)} \\
\phi_{11} &=\scriptsize{ 497664 (3099440 - 852400 w_1 + 698421 w_2 - 80525 w_1 w_2)} \\
\phi_{12} &=\scriptsize{ -13824 (-952134 + 687846 w_1 - 753565 w_2 + 100445 w_1 w_2)} \\
\phi_{13} &=\scriptsize{ 62208 (8344 + 920 w_1 + 519 w_2 - 775 w_1 w_2)} \\
\phi_{14} &=\scriptsize{ -7776 (760 + 104 w_1 - 75 w_2 - 45 w_1 w_2)} \\
\phi_{15} &=\scriptsize{ 864 (36 + 4 w_1 - 23 w_2 + 3 w_1 w_2)} \\
\psi_0 &=\scriptsize{ \frac{2048}{17} (-4136672849 + 1006006169 w_1}\\
&\qquad\qquad\scriptsize{+ 2352737338 w_2 - 571013714 w_1 w_2)} \\
\psi_1 &=\scriptsize{ -256 (414957636 - 109088932 w_1 - 55404821 w_2 + 16225613 w_1 w_2)} \\
\psi_2 &=\scriptsize{ 128 (-61526040 + 11295704 w_1 - 2316397 w_2 + 1783781 w_1 w_2)} \\
\psi_3 &=\scriptsize{ -32 (3493524 - 1924404 w_1 + 2748641 w_2 - 270553 w_1 w_2)} \\
\psi_4 &=\scriptsize{ 64 (-25778 - 5674 w_1 - 7683 w_2 + 8633 w_1 w_2)} \\
\psi_5 &=\scriptsize{ -4 (-2348 + 7980 w_1 - 6541 w_2 + 2357 w_1 w_2)} \\
\psi_6 &=\scriptsize{ 2 (2016 + 496 w_1 + 55 w_2 - 87 w_1 w_2)} \\
\psi_7 &=\scriptsize{ -\frac{1}{2} (68 + 60 w_1 + 17 w_2 - 9 w_1 w_2)}
\end{split}
\right.
\end{gather*}