Elliptic curve isogeny class 34.2-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-34.2-a
• Conductor: 34.2
• Rank: \( 0 \)

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

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

Elliptic curves in class 34.2-a over \(\Q(\sqrt{2}) \)

Isogeny class 34.2-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label	Weierstrass Coefficients
34.2-a1	\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -2 a + 4\) , \( 0\bigr] \)
34.2-a2	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 15 a + 20\) , \( 44 a + 62\bigr] \)
34.2-a3	\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -65 a - 140\) , \( 316 a + 350\bigr] \)
34.2-a4	\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 8 a - 16\) , \( 10 a - 4\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph