Elliptic curve isogeny class 81.1-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-81.1-a
• Conductor: 81.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 81.1-a over \(\Q(\sqrt{2}) \)

Isogeny class 81.1-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label	Weierstrass Coefficients
81.1-a1	\( \bigl[a\) , \( 1\) , \( a + 1\) , \( 358 a - 538\) , \( -4653 a + 6633\bigr] \)
81.1-a2	\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -2 a + 2\) , \( 18 a - 27\bigr] \)
81.1-a3	\( \bigl[a\) , \( 1\) , \( 1\) , \( -17 a - 24\) , \( -52 a - 75\bigr] \)
81.1-a4	\( \bigl[a\) , \( 1\) , \( 1\) , \( -1457 a - 2184\) , \( 36713 a + 52395\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 5 & 10 & 2 \\ 5 & 1 & 2 & 10 \\ 10 & 2 & 1 & 5 \\ 2 & 10 & 5 & 1 \end{array}\right)\)

Isogeny graph