Elliptic curve isogeny class 81.1-c over number field \(\Q(\sqrt{65}) \)

• Base field: \(\Q(\sqrt{65}) \)
• Label: 2.2.65.1-81.1-c
• Conductor: 81.1
• Rank: \( 0 \)

Base field \(\Q(\sqrt{65}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 16 \); class number \(2\).

Elliptic curves in class 81.1-c over \(\Q(\sqrt{65}) \)

Isogeny class 81.1-c contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
81.1-c1	\( \bigl[1\) , \( -1\) , \( 0\) , \( -1452 a - 5130\) , \( 74570 a + 263319\bigr] \)
81.1-c2	\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 214 a + 760\) , \( 5399 a + 19068\bigr] \)
81.1-c3	\( \bigl[1\) , \( -1\) , \( 0\) , \( -102 a - 360\) , \( 932 a + 3291\bigr] \)
81.1-c4	\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -5 a - 8\) , \( -7 a - 4\bigr] \)
81.1-c5	\( \bigl[1\) , \( -1\) , \( 0\) , \( -1542 a - 5445\) , \( 66560 a + 235032\bigr] \)
81.1-c6	\( \bigl[1\) , \( -1\) , \( 0\) , \( -24672 a - 87120\) , \( 4241462 a + 14977149\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph