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

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-1024.1-b
• Conductor: 1024.1
• Rank: \( 1 \)

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

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

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

Isogeny class 1024.1-b contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
1024.1-b1	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 10 a - 1\) , \( -7 a + 15\bigr] \)
1024.1-b2	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -1\) , \( -a - 1\bigr] \)
1024.1-b3	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a - 2\) , \( 4 a + 6\bigr] \)
1024.1-b4	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -10 a - 21\) , \( -39 a - 61\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph