Elliptic curve isogeny class 1568.1-e over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-1568.1-e
• Conductor: 1568.1
• Rank: \( 1 \)

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

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

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph

Elliptic curves in class 1568.1-e over \(\Q(\sqrt{2}) \)

Isogeny class 1568.1-e contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
1568.1-e1	\( \bigl[0\) , \( -1\) , \( 0\) , \( 4 a + 4\) , \( 4 a + 6\bigr] \)
1568.1-e2	\( \bigl[0\) , \( 1\) , \( 0\) , \( -4 a - 10\) , \( 4 a + 8\bigr] \)
1568.1-e3	\( \bigl[a\) , \( 0\) , \( 0\) , \( -a - 20\) , \( -24 a - 6\bigr] \)
1568.1-e4	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 3 a - 7\) , \( -74 a + 105\bigr] \)