Elliptic curve isogeny class 686.1-f over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-686.1-f
• Conductor: 686.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 686.1-f over \(\Q(\sqrt{2}) \)

Isogeny class 686.1-f contains 2 curves linked by isogenies of degree 2.

Curve label	Weierstrass Coefficients
686.1-f1	\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 9 a - 7\) , \( 16 a - 19\bigr] \)
686.1-f2	\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -176 a - 256\) , \( 36 a + 32\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph