Elliptic curve isogeny class 605.1-c over number field \(\Q(\sqrt{5}) \)

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-605.1-c
• Conductor: 605.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 605.1-c over \(\Q(\sqrt{5}) \)

Isogeny class 605.1-c contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
605.1-c1	\( \bigl[\phi\) , \( -\phi - 1\) , \( 0\) , \( 48 \phi - 38\) , \( -1458 \phi - 1109\bigr] \)
605.1-c2	\( \bigl[1\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \)
605.1-c3	\( \bigl[1\) , \( -1\) , \( 0\) , \( -4\) , \( 3\bigr] \)
605.1-c4	\( \bigl[1\) , \( -1\) , \( 0\) , \( -29\) , \( -52\bigr] \)
605.1-c5	\( \bigl[1\) , \( -1\) , \( 0\) , \( -59\) , \( 190\bigr] \)
605.1-c6	\( \bigl[\phi + 1\) , \( 1\) , \( \phi + 1\) , \( -48 \phi + 9\) , \( 1410 \phi - 2558\bigr] \)

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