Elliptic curve isogeny class 784.1-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-784.1-a
• Conductor: 784.1
• Rank: \( 0 \)

Base field \(\Q(\sqrt{2}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 2 \); class number \(1\).

Elliptic curves in class 784.1-a over \(\Q(\sqrt{2}) \)

Isogeny class 784.1-a contains 12 curves linked by isogenies of degrees dividing 36.

Curve label	Weierstrass Coefficients
784.1-a1	\( \bigl[a\) , \( -1\) , \( 0\) , \( -682\) , \( 6990\bigr] \)
784.1-a2	\( \bigl[a\) , \( -1\) , \( 0\) , \( -2\) , \( -2\bigr] \)
784.1-a3	\( \bigl[a\) , \( -1\) , \( 0\) , \( 18\) , \( 46\bigr] \)
784.1-a4	\( \bigl[a\) , \( -1\) , \( 0\) , \( 220 a - 362\) , \( 2320 a - 3330\bigr] \)
784.1-a5	\( \bigl[a\) , \( -1\) , \( 0\) , \( -142\) , \( 558\bigr] \)
784.1-a6	\( \bigl[a\) , \( -1\) , \( 0\) , \( 520 a - 1422\) , \( -16000 a + 16302\bigr] \)
784.1-a7	\( \bigl[a\) , \( -1\) , \( 0\) , \( 57920 a - 92842\) , \( -9795840 a + 14293838\bigr] \)
784.1-a8	\( \bigl[a\) , \( -1\) , \( 0\) , \( -42\) , \( -98\bigr] \)
784.1-a9	\( \bigl[a\) , \( -1\) , \( 0\) , \( -520 a - 1422\) , \( 16000 a + 16302\bigr] \)
784.1-a10	\( \bigl[a\) , \( -1\) , \( 0\) , \( -10922\) , \( 441166\bigr] \)
784.1-a11	\( \bigl[a\) , \( -1\) , \( 0\) , \( -220 a - 362\) , \( -2320 a - 3330\bigr] \)
784.1-a12	\( \bigl[a\) , \( -1\) , \( 0\) , \( -57920 a - 92842\) , \( 9795840 a + 14293838\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrrrr} 1 & 9 & 3 & 36 & 6 & 12 & 4 & 18 & 12 & 2 & 36 & 4 \\ 9 & 1 & 3 & 4 & 6 & 12 & 36 & 2 & 12 & 18 & 4 & 36 \\ 3 & 3 & 1 & 12 & 2 & 4 & 12 & 6 & 4 & 6 & 12 & 12 \\ 36 & 4 & 12 & 1 & 6 & 12 & 36 & 2 & 3 & 18 & 4 & 9 \\ 6 & 6 & 2 & 6 & 1 & 2 & 6 & 3 & 2 & 3 & 6 & 6 \\ 12 & 12 & 4 & 12 & 2 & 1 & 3 & 6 & 4 & 6 & 3 & 12 \\ 4 & 36 & 12 & 36 & 6 & 3 & 1 & 18 & 12 & 2 & 9 & 4 \\ 18 & 2 & 6 & 2 & 3 & 6 & 18 & 1 & 6 & 9 & 2 & 18 \\ 12 & 12 & 4 & 3 & 2 & 4 & 12 & 6 & 1 & 6 & 12 & 3 \\ 2 & 18 & 6 & 18 & 3 & 6 & 2 & 9 & 6 & 1 & 18 & 2 \\ 36 & 4 & 12 & 4 & 6 & 3 & 9 & 2 & 12 & 18 & 1 & 36 \\ 4 & 36 & 12 & 9 & 6 & 12 & 4 & 18 & 3 & 2 & 36 & 1 \end{array}\right)\)

Isogeny graph