Elliptic curve isogeny class 578.3-e over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-578.3-e
• Conductor: 578.3
• Rank: \( 1 \)

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

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

Elliptic curves in class 578.3-e over \(\Q(\sqrt{2}) \)

Isogeny class 578.3-e contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
578.3-e1	\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 90 a - 202\) , \( -632 a + 1139\bigr] \)
578.3-e2	\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -2\) , \( -1\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph