Elliptic curve isogeny class 196.3-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-196.3-a
• Conductor: 196.3
• Rank: \( 0 \)

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

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

Elliptic curves in class 196.3-a over \(\Q(\sqrt{2}) \)

Isogeny class 196.3-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
196.3-a1	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -312 a - 479\) , \( 4555 a + 6314\bigr] \)
196.3-a2	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 8 a - 7\) , \( 7 a - 8\bigr] \)
196.3-a3	\( \bigl[a\) , \( -a\) , \( 0\) , \( 9 a - 15\) , \( -14 a + 20\bigr] \)
196.3-a4	\( \bigl[a\) , \( -a\) , \( 0\) , \( -6 a - 110\) , \( 2170 a - 2612\bigr] \)
196.3-a5	\( \bigl[a\) , \( -a\) , \( 0\) , \( 154 a - 240\) , \( -1218 a + 1770\bigr] \)
196.3-a6	\( \bigl[a\) , \( -a\) , \( 0\) , \( -16 a - 10\) , \( -98 a + 76\bigr] \)
196.3-a7	\( \bigl[a\) , \( -a\) , \( 0\) , \( 79 a - 225\) , \( 1092 a - 1044\bigr] \)
196.3-a8	\( \bigl[a\) , \( -a\) , \( 0\) , \( -436 a - 1200\) , \( 10318 a + 15770\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 6 & 4 & 12 & 12 & 2 & 4 \\ 3 & 1 & 2 & 12 & 4 & 4 & 6 & 12 \\ 6 & 2 & 1 & 6 & 2 & 2 & 3 & 6 \\ 4 & 12 & 6 & 1 & 12 & 3 & 2 & 4 \\ 12 & 4 & 2 & 12 & 1 & 4 & 6 & 3 \\ 12 & 4 & 2 & 3 & 4 & 1 & 6 & 12 \\ 2 & 6 & 3 & 2 & 6 & 6 & 1 & 2 \\ 4 & 12 & 6 & 4 & 3 & 12 & 2 & 1 \end{array}\right)\)

Isogeny graph