Elliptic curve isogeny class 3136.3-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-3136.3-a
• Conductor: 3136.3
• Rank: \( 1 \)

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

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

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph

Elliptic curves in class 3136.3-a over \(\Q(\sqrt{2}) \)

Isogeny class 3136.3-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
3136.3-a1	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 19 a + 25\) , \( 135 a + 185\bigr] \)
3136.3-a2	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 21 a + 26\) , \( -115 a - 160\bigr] \)
3136.3-a3	\( \bigl[0\) , \( -1\) , \( 0\) , \( 10 a - 18\) , \( 20 a - 24\bigr] \)
3136.3-a4	\( \bigl[0\) , \( 1\) , \( 0\) , \( 10 a - 18\) , \( -20 a + 24\bigr] \)
3136.3-a5	\( \bigl[a\) , \( -1\) , \( a\) , \( -10 a - 3\) , \( 25 a - 8\bigr] \)
3136.3-a6	\( \bigl[a\) , \( 0\) , \( a\) , \( -10 a - 3\) , \( -25 a + 7\bigr] \)