Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-55.1-a
Conductor 55.1
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 55.1-a over \(\Q(\sqrt{5}) \)

Isogeny class 55.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
55.1-a1 \( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( 9 \phi - 25\) , \( -6 \phi + 44\bigr] \)
55.1-a2 \( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( 54 \phi\) , \( -374 \phi - 198\bigr] \)
55.1-a3 \( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -\phi - 1\) , \( -\phi\bigr] \)
55.1-a4 \( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( -21 \phi - 25\) , \( -54 \phi - 58\bigr] \)
55.1-a5 \( \bigl[1\) , \( -\phi + 1\) , \( 1\) , \( -16 \phi - 210\) , \( 1110 \phi - 534\bigr] \)
55.1-a6 \( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( 4 \phi - 11\) , \( -9 \phi + 13\bigr] \)
55.1-a7 \( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -6 \phi - 1\) , \( \phi - 17\bigr] \)
55.1-a8 \( \bigl[\phi + 1\) , \( 0\) , \( \phi + 1\) , \( -6 \phi - 26\) , \( 28 \phi + 8\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 6 & 12 & 2 & 12 & 4 \\ 3 & 1 & 12 & 2 & 4 & 6 & 4 & 12 \\ 4 & 12 & 1 & 6 & 12 & 2 & 3 & 4 \\ 6 & 2 & 6 & 1 & 2 & 3 & 2 & 6 \\ 12 & 4 & 12 & 2 & 1 & 6 & 4 & 3 \\ 2 & 6 & 2 & 3 & 6 & 1 & 6 & 2 \\ 12 & 4 & 3 & 2 & 4 & 6 & 1 & 12 \\ 4 & 12 & 4 & 6 & 3 & 2 & 12 & 1 \end{array}\right)\)

Isogeny graph