Elliptic curve isogeny class 578.1-d over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-578.1-d
• Conductor: 578.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 578.1-d over \(\Q(\sqrt{2}) \)

Isogeny class 578.1-d contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
578.1-d1	\( \bigl[1\) , \( 0\) , \( 0\) , \( 230 a - 363\) , \( -2390 a + 3513\bigr] \)
578.1-d2	\( \bigl[1\) , \( 0\) , \( 0\) , \( 355 a - 193\) , \( -3135 a + 2315\bigr] \)
578.1-d3	\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \)
578.1-d4	\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \)
578.1-d5	\( \bigl[1\) , \( 0\) , \( 0\) , \( -355 a - 193\) , \( 3135 a + 2315\bigr] \)
578.1-d6	\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \)
578.1-d7	\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \)
578.1-d8	\( \bigl[1\) , \( 0\) , \( 0\) , \( -230 a - 363\) , \( 2390 a + 3513\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 6 & 12 & 2 & 12 & 4 \\ 3 & 1 & 12 & 2 & 4 & 6 & 4 & 12 \\ 4 & 12 & 1 & 6 & 12 & 2 & 3 & 4 \\ 6 & 2 & 6 & 1 & 2 & 3 & 2 & 6 \\ 12 & 4 & 12 & 2 & 1 & 6 & 4 & 3 \\ 2 & 6 & 2 & 3 & 6 & 1 & 6 & 2 \\ 12 & 4 & 3 & 2 & 4 & 6 & 1 & 12 \\ 4 & 12 & 4 & 6 & 3 & 2 & 12 & 1 \end{array}\right)\)

Isogeny graph