Elliptic curve isogeny class 4096.1-x over number field \(\Q(\sqrt{5}) \)

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-4096.1-x
• Conductor: 4096.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 4096.1-x over \(\Q(\sqrt{5}) \)

Isogeny class 4096.1-x contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
4096.1-x1	\( \bigl[0\) , \( -1\) , \( 0\) , \( 12 \phi - 45\) , \( -932 \phi - 411\bigr] \)
4096.1-x2	\( \bigl[0\) , \( \phi + 1\) , \( 0\) , \( 21 \phi - 22\) , \( 44 \phi - 57\bigr] \)
4096.1-x3	\( \bigl[0\) , \( \phi + 1\) , \( 0\) , \( \phi - 2\) , \( -1\bigr] \)
4096.1-x4	\( \bigl[0\) , \( \phi + 1\) , \( 0\) , \( \phi - 22\) , \( -44 \phi + 11\bigr] \)
4096.1-x5	\( \bigl[0\) , \( \phi + 1\) , \( 0\) , \( -19 \phi - 2\) , \( 24 \phi + 11\bigr] \)
4096.1-x6	\( \bigl[0\) , \( 1\) , \( 0\) , \( -12 \phi - 33\) , \( -932 \phi + 1343\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 8 & 4 \\ 8 & 1 & 2 & 4 & 4 & 8 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 2 & 4 & 2 & 1 & 4 & 2 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 8 & 4 & 2 & 8 & 1 \end{array}\right)\)

Isogeny graph