Elliptic curve isogeny class 1024.1-d over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-1024.1-d
• Number of curves: 6
• Conductor: 1024.1
• Rank: \( 1 \)

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

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

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 2 & 8 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 2 & 2 & 1 & 4 & 2 \\ 4 & 8 & 2 & 4 & 1 & 8 \\ 8 & 4 & 4 & 2 & 8 & 1 \end{array}\right)\)

Isogeny graph

Elliptic curves in class 1024.1-d over \(\Q(\sqrt{2}) \)

Isogeny class 1024.1-d contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
1024.1-d1	\( \bigl[0\) , \( -1\) , \( 0\) , \( 20 a - 23\) , \( 40 a - 69\bigr] \)
1024.1-d2	\( \bigl[0\) , \( 1\) , \( 0\) , \( 20 a - 23\) , \( -40 a + 69\bigr] \)
1024.1-d3	\( \bigl[0\) , \( -1\) , \( 0\) , \( -3\) , \( -1\bigr] \)
1024.1-d4	\( \bigl[0\) , \( 1\) , \( 0\) , \( -3\) , \( 1\bigr] \)
1024.1-d5	\( \bigl[0\) , \( -1\) , \( 0\) , \( -20 a - 23\) , \( -40 a - 69\bigr] \)
1024.1-d6	\( \bigl[0\) , \( 1\) , \( 0\) , \( -20 a - 23\) , \( 40 a + 69\bigr] \)