Elliptic curve isogeny class 686.1-d over number field Q(2)\Q(\sqrt{2})

• Base field: $\Q(\sqrt{2})$
• Label: 2.2.8.1-686.1-d
• Conductor: 686.1
• Rank: $0$

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

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

Elliptic curves in class 686.1-d over $\Q(\sqrt{2})$

Isogeny class 686.1-d contains 12 curves linked by isogenies of degrees dividing 36.

Curve label	Weierstrass Coefficients
686.1-d1	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-683 a - 1536$ , $19222 a + 21843\bigr]$
686.1-d2	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-3 a - 6$ , $-6 a - 7\bigr]$
686.1-d3	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $17 a + 39$ , $126 a + 143\bigr]$
686.1-d4	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $132 a - 376$ , $-1908 a + 2353\bigr]$
686.1-d5	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-143 a - 321$ , $1534 a + 1743\bigr]$
686.1-d6	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-253 a - 2161$ , $-5170 a - 37057\bigr]$
686.1-d7	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $37477 a - 93056$ , $8696054 a - 9208877\bigr]$
686.1-d8	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-43 a - 96$ , $-270 a - 307\bigr]$
686.1-d9	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-2593 a - 4241$ , $94830 a + 138943\bigr]$
686.1-d10	$\bigl[a + 1$ , $-a - 1$ , $a + 1$ , $-10923 a - 24576$ , $1213206 a + 1378643\bigr]$
686.1-d11	$\bigl[1$ , $-a + 1$ , $1$ , $485 a - 759$ , $-2705 a + 3591\bigr]$
686.1-d12	$\bigl[1$ , $-a + 1$ , $1$ , $103060 a - 164599$ , $23909933 a - 32816619\bigr]$

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