Properties

Base field 4.4.9909.1
Label 4.4.9909.1-21.1-d
Conductor 21.1
Rank not recorded

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Base field 4.4.9909.1

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

Elliptic curves in class 21.1-d over 4.4.9909.1

Isogeny class 21.1-d contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
21.1-d1 \( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 87 a^{3} - 17 a^{2} - 418 a - 327\) , \( -740 a^{3} - 97 a^{2} + 3864 a + 3503\bigr] \)
21.1-d2 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -544 a^{3} + 506 a^{2} + 2629 a - 1404\) , \( -13092 a^{3} + 14266 a^{2} + 61674 a - 33259\bigr] \)
21.1-d3 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -84 a^{3} + 146 a^{2} + 299 a - 174\) , \( -188 a^{3} + 858 a^{2} + 1498 a - 961\bigr] \)
21.1-d4 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -34 a^{3} + 6 a^{2} + 144 a - 69\) , \( -240 a^{3} + 410 a^{2} + 1244 a - 706\bigr] \)
21.1-d5 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( a^{3} - 4 a^{2} - 6 a + 6\) , \( a^{2} - a - 1\bigr] \)
21.1-d6 \( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( 36 a^{3} - 554 a^{2} - 426 a + 351\) , \( 660 a^{3} + 9720 a^{2} + 6098 a - 5674\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 4 & 4 & 2 & 2 & 2 \\ 4 & 1 & 4 & 2 & 8 & 8 \\ 4 & 4 & 1 & 2 & 8 & 8 \\ 2 & 2 & 2 & 1 & 4 & 4 \\ 2 & 8 & 8 & 4 & 1 & 4 \\ 2 & 8 & 8 & 4 & 4 & 1 \end{array}\right)\)

Isogeny graph