Elliptic curve isogeny class 578.1-f over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-578.1-f
• Conductor: 578.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 578.1-f over \(\Q(\sqrt{2}) \)

Isogeny class 578.1-f contains 4 curves linked by isogenies of degrees dividing 6.

Curve label	Weierstrass Coefficients
578.1-f1	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 17 a - 32\) , \( -59 a + 71\bigr] \)
578.1-f2	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 7 a - 12\) , \( 9 a - 13\bigr] \)
578.1-f3	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -445 a - 685\) , \( -6530 a - 9359\bigr] \)
578.1-f4	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 462 a - 1272\) , \( 8745 a - 18675\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 6 & 3 \\ 2 & 1 & 3 & 6 \\ 6 & 3 & 1 & 2 \\ 3 & 6 & 2 & 1 \end{array}\right)\)

Isogeny graph