Properties

Base field 4.4.19821.1
Label 4.4.19821.1-21.1-a
Conductor 21.1
Rank not recorded

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Base field 4.4.19821.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).

Elliptic curves in class 21.1-a over 4.4.19821.1

Isogeny class 21.1-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
21.1-a1 \( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 2\) , \( a^{2} - a - 3\) , \( -\frac{1}{3} a^{3} - \frac{4}{3} a^{2} + 1\) , \( \frac{2}{3} a^{3} + \frac{2}{3} a^{2} - 2 a - 2\bigr] \)
21.1-a2 \( \bigl[\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a\) , \( \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 2\) , \( a^{2} - a - 3\) , \( -\frac{26}{3} a^{3} - \frac{44}{3} a^{2} + 25 a + 1\) , \( 47 a^{3} + 84 a^{2} - 120 a - 36\bigr] \)
21.1-a3 \( \bigl[a + 1\) , \( -a^{2} + 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( \frac{28}{3} a^{3} + \frac{1}{3} a^{2} - 75 a - 21\) , \( 37 a^{3} + 4 a^{2} - 292 a - 103\bigr] \)
21.1-a4 \( \bigl[a + 1\) , \( -a^{2} + 3\) , \( -\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\) , \( -19 a^{3} - 3 a^{2} + 150 a + 54\) , \( 148 a^{3} + 16 a^{2} - 1167 a - 407\bigr] \)

Rank

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Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph