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

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-686.2-e
• Conductor: 686.2
• Rank: \( 0 \)

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

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

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

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

Curve label	Weierstrass Coefficients
686.2-e1	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 478 a - 695\) , \( 6733 a - 9853\bigr] \)
686.2-e2	\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 8 a - 5\) , \( 15 a - 5\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph