Elliptic curve isogeny class 16.5-b over number field \(\Q(\sqrt{65}) \)

• Base field: \(\Q(\sqrt{65}) \)
• Label: 2.2.65.1-16.5-b
• Conductor: 16.5
• Rank: \( 1 \)

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

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

Elliptic curves in class 16.5-b over \(\Q(\sqrt{65}) \)

Isogeny class 16.5-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
16.5-b1	\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 775 a - 3498\) , \( 23659 a - 107193\bigr] \)
16.5-b2	\( \bigl[0\) , \( 1\) , \( 0\) , \( 16 a - 72\) , \( -60 a + 272\bigr] \)
16.5-b3	\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 5 a - 8\) , \( 7 a - 21\bigr] \)
16.5-b4	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -1359 a + 6168\) , \( 84736 a - 383956\bigr] \)
16.5-b5	\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 50 a - 213\) , \( 378 a - 1704\bigr] \)
16.5-b6	\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 45 a - 208\) , \( 421 a - 1907\bigr] \)

Rank

Rank: \( 1 \)

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