Elliptic curve isogeny class 209.3-b over number field \(\Q(\sqrt{5}) \)

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-209.3-b
• Conductor: 209.3
• Rank: \( 0 \)

Base field \(\Q(\sqrt{5}) \)

Generator \(\phi\), with minimal polynomial \( x^{2} - x - 1 \); class number \(1\).

Elliptic curves in class 209.3-b over \(\Q(\sqrt{5}) \)

Isogeny class 209.3-b contains 4 curves linked by isogenies of degrees dividing 6.

Curve label	Weierstrass Coefficients
209.3-b1	\( \bigl[1\) , \( 0\) , \( \phi\) , \( -8 \phi - 3\) , \( -5 \phi - 6\bigr] \)
209.3-b2	\( \bigl[1\) , \( 0\) , \( \phi\) , \( -3 \phi - 3\) , \( 3 \phi + 1\bigr] \)
209.3-b3	\( \bigl[1\) , \( 0\) , \( \phi\) , \( 22 \phi - 103\) , \( 127 \phi - 418\bigr] \)
209.3-b4	\( \bigl[\phi + 1\) , \( -\phi\) , \( \phi\) , \( 288 \phi - 623\) , \( 3651 \phi - 6655\bigr] \)

Rank

Rank: \( 0 \)

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