Elliptic curve isogeny class 31.2-a over number field \(\Q(\sqrt{5}) \)

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-31.2-a
• Conductor: 31.2
• Rank: \( 0 \)

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

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

Elliptic curves in class 31.2-a over \(\Q(\sqrt{5}) \)

Isogeny class 31.2-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
31.2-a1	\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( -30 \phi - 45\) , \( -111 \phi - 117\bigr] \)
31.2-a2	\( \bigl[\phi\) , \( 1\) , \( \phi + 1\) , \( 16 \phi - 2\) , \( -172 \phi - 94\bigr] \)
31.2-a3	\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( 0\) , \( 0\bigr] \)
31.2-a4	\( \bigl[1\) , \( -\phi - 1\) , \( \phi\) , \( -5\) , \( -3 \phi + 3\bigr] \)
31.2-a5	\( \bigl[\phi + 1\) , \( -\phi - 1\) , \( \phi\) , \( 10 \phi - 32\) , \( -43 \phi + 53\bigr] \)
31.2-a6	\( \bigl[\phi + 1\) , \( -\phi - 1\) , \( \phi\) , \( -135 \phi - 7\) , \( -738 \phi - 26\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph