Elliptic curve isogeny class 1024.1-e over number field \(\Q(\sqrt{5}) \)

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-1024.1-e
• Conductor: 1024.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 1024.1-e over \(\Q(\sqrt{5}) \)

Isogeny class 1024.1-e contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
1024.1-e1	\( \bigl[0\) , \( -\phi - 1\) , \( 0\) , \( 9 \phi - 1\) , \( \phi + 8\bigr] \)
1024.1-e2	\( \bigl[0\) , \( -1\) , \( 0\) , \( \phi - 2\) , \( -\phi + 2\bigr] \)
1024.1-e3	\( \bigl[0\) , \( -1\) , \( 0\) , \( -4 \phi + 3\) , \( -4 \phi + 5\bigr] \)
1024.1-e4	\( \bigl[0\) , \( -1\) , \( 0\) , \( \phi - 12\) , \( 13 \phi - 4\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph