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

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-1024.1-e
• 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-e over \(\Q(\sqrt{2}) \)

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

Curve label	Weierstrass Coefficients
1024.1-e1	\( \bigl[0\) , \( a\) , \( 0\) , \( 24 a + 32\) , \( 32 a + 48\bigr] \)
1024.1-e2	\( \bigl[0\) , \( -1\) , \( 0\) , \( 2 a - 3\) , \( -2 a + 3\bigr] \)
1024.1-e3	\( \bigl[0\) , \( 1\) , \( 0\) , \( -2 a + 2\) , \( -2 a + 2\bigr] \)
1024.1-e4	\( \bigl[0\) , \( -1\) , \( 0\) , \( 12 a - 23\) , \( 32 a - 37\bigr] \)

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