Elliptic curve isogeny class 17.1-a over number field \(\Q(\sqrt{85}) \)

• Base field: \(\Q(\sqrt{85}) \)
• Label: 2.2.85.1-17.1-a
• Conductor: 17.1
• Rank: \( 2 \)

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

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

Elliptic curves in class 17.1-a over \(\Q(\sqrt{85}) \)

Isogeny class 17.1-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
17.1-a1	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 9 a - 39\) , \( 1563 a - 7983\bigr] \)
17.1-a2	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 9 a - 39\) , \( -5 a + 39\bigr] \)
17.1-a3	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 64 a - 324\) , \( 608 a - 3108\bigr] \)
17.1-a4	\( \bigl[a + 1\) , \( -1\) , \( a\) , \( 999 a - 5169\) , \( 37685 a - 192981\bigr] \)

Rank

Rank: \( 2 \)

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