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

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

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

Curve label	Weierstrass Coefficients
686.1-e1	\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -476 a - 696\) , \( -7429 a - 10807\bigr] \)
686.1-e2	\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -6 a - 6\) , \( -21 a - 19\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph