Elliptic curve isogeny class 45.1-a over number field \(\Q(\sqrt{5}) \)

• Base field: \(\Q(\sqrt{5}) \)
• Label: 2.2.5.1-45.1-a
• Conductor: 45.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 45.1-a over \(\Q(\sqrt{5}) \)

Isogeny class 45.1-a contains 10 curves linked by isogenies of degrees dividing 32.

Curve label	Weierstrass Coefficients
45.1-a1	\( \bigl[\phi\) , \( \phi + 1\) , \( 1\) , \( -4364 \phi - 7739\) , \( -255406 \phi - 296465\bigr] \)
45.1-a2	\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \)
45.1-a3	\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \)
45.1-a4	\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \)
45.1-a5	\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \)
45.1-a6	\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \)
45.1-a7	\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \)
45.1-a8	\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \)
45.1-a9	\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \)
45.1-a10	\( \bigl[\phi + 1\) , \( \phi - 1\) , \( \phi + 1\) , \( 4364 \phi - 12105\) , \( 243301 \phi - 535402\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrr} 1 & 8 & 32 & 16 & 8 & 16 & 4 & 32 & 2 & 4 \\ 8 & 1 & 16 & 8 & 4 & 8 & 2 & 16 & 4 & 8 \\ 32 & 16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 & 32 \\ 16 & 8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 & 16 \\ 8 & 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 & 8 \\ 16 & 8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 & 16 \\ 4 & 2 & 8 & 4 & 2 & 4 & 1 & 8 & 2 & 4 \\ 32 & 16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 & 32 \\ 2 & 4 & 16 & 8 & 4 & 8 & 2 & 16 & 1 & 2 \\ 4 & 8 & 32 & 16 & 8 & 16 & 4 & 32 & 2 & 1 \end{array}\right)\)

Isogeny graph