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

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-686.1-a
• 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-a over \(\Q(\sqrt{2}) \)

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

Curve label	Weierstrass Coefficients
686.1-a1	\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 43 a - 6\) , \( 65 a - 135\bigr] \)
686.1-a2	\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 303 a - 646\) , \( 4929 a - 6831\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph