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

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

Isogeny class 686.1-g contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
686.1-g1	\( \bigl[1\) , \( -a\) , \( a\) , \( 13 a - 39\) , \( -67 a + 51\bigr] \)
686.1-g2	\( \bigl[1\) , \( -a\) , \( a\) , \( -37 a + 216\) , \( -64 a + 756\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph