Elliptic curve isogeny class 128.1-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-128.1-a
• Conductor: 128.1
• Rank: \( 0 \)

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

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

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph

Elliptic curves in class 128.1-a over \(\Q(\sqrt{2}) \)

Isogeny class 128.1-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
128.1-a1	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 2 a + 3\) , \( -5 a - 7\bigr] \)
128.1-a2	\( \bigl[a\) , \( -a\) , \( a\) , \( -4 a - 7\) , \( 6 a + 8\bigr] \)
128.1-a3	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 4 a - 6\) , \( -10 a + 14\bigr] \)
128.1-a4	\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 4 a - 7\) , \( 6 a - 9\bigr] \)