Elliptic curve isogeny class 31.2-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-31.2-a
• Conductor: 31.2
• Rank: \( 0 \)

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

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

Elliptic curves in class 31.2-a over \(\Q(\sqrt{2}) \)

Isogeny class 31.2-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
31.2-a1	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 0\) , \( 0\bigr] \)
31.2-a2	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -25 a - 49\) , \( 79 a + 100\bigr] \)
31.2-a3	\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -10 a - 14\) , \( -21 a - 30\bigr] \)
31.2-a4	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 5 a - 20\) , \( 10 a - 40\bigr] \)

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