Elliptic curve isogeny class 686.2-b over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-686.2-b
• Conductor: 686.2
• Rank: \( 1 \)

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

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

Elliptic curves in class 686.2-b over \(\Q(\sqrt{2}) \)

Isogeny class 686.2-b contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
686.2-b1	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -6 a - 8\) , \( 6 a + 10\bigr] \)
686.2-b2	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 24 a + 47\) , \( 66 a + 71\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph