Elliptic curve isogeny class 9.2-a over number field \(\Q(\sqrt{85}) \)

• Base field: \(\Q(\sqrt{85}) \)
• Label: 2.2.85.1-9.2-a
• Conductor: 9.2
• Rank: \( 1 \)

Base field \(\Q(\sqrt{85}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 21 \); class number \(2\).

Elliptic curves in class 9.2-a over \(\Q(\sqrt{85}) \)

Isogeny class 9.2-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label	Weierstrass Coefficients
9.2-a1	\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 6 a - 26\) , \( 24 a - 113\bigr] \)
9.2-a2	\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( a + 4\) , \( -5\bigr] \)
9.2-a3	\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -177 a - 716\) , \( 2528 a + 10395\bigr] \)
9.2-a4	\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( -2812 a - 11546\) , \( 171148 a + 703383\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph