Elliptic curve isogeny class 256.1-c over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-256.1-c
• Conductor: 256.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 256.1-c over \(\Q(\sqrt{2}) \)

Isogeny class 256.1-c contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
256.1-c1	\( \bigl[0\) , \( -a\) , \( 0\) , \( 10 a - 11\) , \( 23 a - 30\bigr] \)
256.1-c2	\( \bigl[0\) , \( a\) , \( 0\) , \( 10 a - 11\) , \( -23 a + 30\bigr] \)
256.1-c3	\( \bigl[0\) , \( -a\) , \( 0\) , \( -1\) , \( a\bigr] \)
256.1-c4	\( \bigl[0\) , \( a\) , \( 0\) , \( -1\) , \( -a\bigr] \)
256.1-c5	\( \bigl[0\) , \( a\) , \( 0\) , \( -10 a - 11\) , \( -23 a - 30\bigr] \)
256.1-c6	\( \bigl[0\) , \( -a\) , \( 0\) , \( -10 a - 11\) , \( 23 a + 30\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph