Elliptic curve isogeny class 3.1-b over number field \(\Q(\sqrt{85}) \)

• Base field: \(\Q(\sqrt{85}) \)
• Label: 2.2.85.1-3.1-b
• Conductor: 3.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 3.1-b over \(\Q(\sqrt{85}) \)

Isogeny class 3.1-b contains 4 curves linked by isogenies of degrees dividing 6.

Curve label	Weierstrass Coefficients
3.1-b1	\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( a + 2\bigr] \)
3.1-b2	\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -4 a - 19\) , \( -29 a - 121\bigr] \)
3.1-b3	\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -24 a - 119\) , \( -245 a - 1051\bigr] \)
3.1-b4	\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -434 a - 1804\) , \( -12042 a - 49534\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph