Elliptic curve isogeny class 31.1-a over number field \(\Q(\sqrt{5}) \)

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-31.1-a
• Conductor: 31.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 31.1-a over \(\Q(\sqrt{5}) \)

Isogeny class 31.1-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
31.1-a1	\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi\) , \( 0\bigr] \)
31.1-a2	\( \bigl[\phi\) , \( -1\) , \( \phi + 1\) , \( 133 \phi - 141\) , \( 737 \phi - 764\bigr] \)
31.1-a3	\( \bigl[\phi\) , \( -1\) , \( \phi + 1\) , \( -12 \phi - 21\) , \( 42 \phi + 10\bigr] \)
31.1-a4	\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( 31 \phi - 75\) , \( 141 \phi - 303\bigr] \)
31.1-a5	\( \bigl[1\) , \( \phi + 1\) , \( \phi\) , \( \phi - 5\) , \( 3 \phi - 5\bigr] \)
31.1-a6	\( \bigl[\phi + 1\) , \( -\phi + 1\) , \( \phi\) , \( -18 \phi + 15\) , \( 171 \phi - 265\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph