Properties

Base field 4.4.4752.1
Label 4.4.4752.1-39.1-f
Conductor 39.1
Rank \( 1 \)

Related objects

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Base field 4.4.4752.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1 \); class number \(1\).

Elliptic curves in class 39.1-f over 4.4.4752.1

Isogeny class 39.1-f contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
39.1-f1 \( \bigl[a^{2} - a - 2\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{2} - 2\) , \( -1385 a^{3} - 686 a^{2} + 2441 a + 526\) , \( -40165 a^{3} - 19914 a^{2} + 70815 a + 16169\bigr] \)
39.1-f2 \( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{3} - 3 a - 1\) , \( 3 a^{2} + 16 a - 90\) , \( -23 a^{3} + 128 a^{2} - 125 a - 175\bigr] \)
39.1-f3 \( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{3} - 3 a - 1\) , \( -7 a^{2} + 31 a - 45\) , \( -5 a^{3} + 39 a^{2} - 113 a + 104\bigr] \)
39.1-f4 \( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a\) , \( a^{3} - 3 a - 1\) , \( -2 a^{2} + a\) , \( -2 a^{2} - 2 a + 5\bigr] \)

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