Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-81.1-a
Conductor 81.1
Rank \( 0 \)

Related objects

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

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

Elliptic curves in class 81.1-a over \(\Q(\sqrt{5}) \)

Isogeny class 81.1-a contains 8 curves linked by isogenies of degrees dividing 30.

Curve label Weierstrass Coefficients
81.1-a1 \( \bigl[1\) , \( -1\) , \( \phi\) , \( -2 \phi\) , \( \phi\bigr] \)
81.1-a2 \( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( -14 \phi - 2\) , \( -21 \phi - 6\bigr] \)
81.1-a3 \( \bigl[1\) , \( -1\) , \( \phi + 1\) , \( \phi - 2\) , \( -2 \phi + 1\bigr] \)
81.1-a4 \( \bigl[1\) , \( -1\) , \( \phi\) , \( 13 \phi - 15\) , \( 20 \phi - 26\bigr] \)
81.1-a5 \( \bigl[\phi\) , \( -\phi - 1\) , \( \phi + 1\) , \( -13 \phi - 14\) , \( -20 \phi - 6\bigr] \)
81.1-a6 \( \bigl[\phi\) , \( -\phi - 1\) , \( \phi\) , \( -113 \phi - 125\) , \( 867 \phi + 384\bigr] \)
81.1-a7 \( \bigl[\phi + 1\) , \( 1\) , \( 1\) , \( 13 \phi - 26\) , \( 32 \phi - 51\bigr] \)
81.1-a8 \( \bigl[\phi + 1\) , \( 1\) , \( 0\) , \( 114 \phi - 237\) , \( -754 \phi + 1014\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 15 & 5 & 3 & 10 & 6 & 2 & 30 \\ 15 & 1 & 3 & 5 & 6 & 10 & 30 & 2 \\ 5 & 3 & 1 & 15 & 2 & 30 & 10 & 6 \\ 3 & 5 & 15 & 1 & 30 & 2 & 6 & 10 \\ 10 & 6 & 2 & 30 & 1 & 15 & 5 & 3 \\ 6 & 10 & 30 & 2 & 15 & 1 & 3 & 5 \\ 2 & 30 & 10 & 6 & 5 & 3 & 1 & 15 \\ 30 & 2 & 6 & 10 & 3 & 5 & 15 & 1 \end{array}\right)\)

Isogeny graph