Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-55.2-a
Conductor 55.2
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 55.2-a over \(\Q(\sqrt{5}) \)

Isogeny class 55.2-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
55.2-a1 \( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( -\phi\) , \( 0\bigr] \)
55.2-a2 \( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( 4 \phi - 5\) , \( -2 \phi - 15\bigr] \)
55.2-a3 \( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( 4 \phi - 30\) , \( -29 \phi + 37\bigr] \)
55.2-a4 \( \bigl[1\) , \( \phi\) , \( 1\) , \( -54 \phi + 54\) , \( 374 \phi - 572\bigr] \)
55.2-a5 \( \bigl[1\) , \( \phi\) , \( 1\) , \( 21 \phi - 46\) , \( 54 \phi - 112\bigr] \)
55.2-a6 \( \bigl[\phi\) , \( -\phi + 1\) , \( \phi\) , \( -6 \phi - 5\) , \( 8 \phi + 5\bigr] \)
55.2-a7 \( \bigl[1\) , \( \phi\) , \( 1\) , \( 16 \phi - 226\) , \( -1110 \phi + 576\bigr] \)
55.2-a8 \( \bigl[1\) , \( \phi\) , \( 1\) , \( -9 \phi - 16\) , \( 6 \phi + 38\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph