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

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-4096.1-k
• 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-k over \(\Q(\sqrt{5}) \)

Isogeny class 4096.1-k contains 4 curves linked by isogenies of degrees dividing 10.

Curve label	Weierstrass Coefficients
4096.1-k1	\( \bigl[0\) , \( -\phi\) , \( 0\) , \( -\phi - 9\) , \( -6 \phi - 15\bigr] \)
4096.1-k2	\( \bigl[0\) , \( \phi\) , \( 0\) , \( -\phi - 9\) , \( 6 \phi + 15\bigr] \)
4096.1-k3	\( \bigl[0\) , \( \phi - 1\) , \( 0\) , \( -2\) , \( -2 \phi + 4\bigr] \)
4096.1-k4	\( \bigl[0\) , \( -\phi + 1\) , \( 0\) , \( -2\) , \( 2 \phi - 4\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 5 & 10 & 2 \\ 5 & 1 & 2 & 10 \\ 10 & 2 & 1 & 5 \\ 2 & 10 & 5 & 1 \end{array}\right)\)

Isogeny graph