Elliptic curve isogeny class 32.1-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-32.1-a
• Conductor: 32.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 32.1-a over \(\Q(\sqrt{2}) \)

Isogeny class 32.1-a contains 8 curves linked by isogenies of degrees dividing 16.

Curve label	Weierstrass Coefficients
32.1-a1	\( \bigl[a\) , \( 1\) , \( 0\) , \( 15 a - 22\) , \( 46 a - 69\bigr] \)
32.1-a2	\( \bigl[a\) , \( 1\) , \( a\) , \( 15 a - 23\) , \( -31 a + 46\bigr] \)
32.1-a3	\( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \)
32.1-a4	\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \)
32.1-a5	\( \bigl[a\) , \( 1\) , \( 0\) , \( -2\) , \( -3\bigr] \)
32.1-a6	\( \bigl[a\) , \( 1\) , \( a\) , \( -3\) , \( 0\bigr] \)
32.1-a7	\( \bigl[a\) , \( 1\) , \( 0\) , \( -15 a - 22\) , \( -46 a - 69\bigr] \)
32.1-a8	\( \bigl[a\) , \( 1\) , \( a\) , \( -15 a - 23\) , \( 31 a + 46\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 8 & 4 & 2 & 8 & 4 & 16 \\ 16 & 1 & 8 & 4 & 8 & 2 & 16 & 4 \\ 8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 2 & 8 & 4 & 2 & 1 & 4 & 2 & 8 \\ 8 & 2 & 4 & 2 & 4 & 1 & 8 & 2 \\ 4 & 16 & 8 & 4 & 2 & 8 & 1 & 16 \\ 16 & 4 & 8 & 4 & 8 & 2 & 16 & 1 \end{array}\right)\)

Isogeny graph