Properties

Base field \(\Q(\zeta_{20})^+\)
Label 4.4.2000.1-320.1-f
Conductor 320.1
Rank \( 1 \)

Related objects

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Base field \(\Q(\zeta_{20})^+\)

Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 5 \); class number \(1\).

Elliptic curves in class 320.1-f over \(\Q(\zeta_{20})^+\)

Isogeny class 320.1-f contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
320.1-f1 \( \bigl[a^{2} + a - 3\) , \( a^{3} + a^{2} - 2 a - 2\) , \( a^{3} + a^{2} - 3 a - 2\) , \( -22 a^{3} - 28 a^{2} + 43 a + 21\) , \( 58 a^{3} + 72 a^{2} - 104 a - 70\bigr] \)
320.1-f2 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - a - 4\) , \( a^{2} + a - 3\) , \( 24 a^{3} - 17 a^{2} - 56 a - 3\) , \( -35 a^{3} + 124 a^{2} + 7 a - 221\bigr] \)
320.1-f3 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{2} - a - 4\) , \( a^{2} + a - 3\) , \( 4 a^{3} - 2 a^{2} - 11 a - 3\) , \( -2 a^{3} - 12 a^{2} + 14 a + 29\bigr] \)
320.1-f4 \( \bigl[0\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 0\) , \( -2 a^{3} + 3 a^{2} + 4 a - 5\) , \( 3 a^{3} - 5 a^{2} - 6 a + 9\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph