Properties

Base field 4.4.2777.1
Label 4.4.2777.1-22.1-a
Conductor 22.1
Rank not recorded

Related objects

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Base field 4.4.2777.1

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

Elliptic curves in class 22.1-a over 4.4.2777.1

Isogeny class 22.1-a contains 12 curves linked by isogenies of degrees dividing 24.

Curve label Weierstrass Coefficients
22.1-a1 \( \bigl[a^{3} - 3 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -a^{3} - 2 a^{2} + 4 a + 4\) , \( -a^{3} + 4 a\bigr] \)
22.1-a2 \( \bigl[a^{3} - 3 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 4 a^{3} - 22 a^{2} - a + 14\) , \( 6 a^{3} - 65 a^{2} + 6 a + 36\bigr] \)
22.1-a3 \( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( -20 a^{3} + 35 a^{2} + 56 a - 64\) , \( 299 a^{3} - 501 a^{2} - 856 a + 876\bigr] \)
22.1-a4 \( \bigl[a^{2} - a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{2} - a - 2\) , \( 255 a^{3} - 345 a^{2} - 799 a + 451\) , \( -6627 a^{3} + 11701 a^{2} + 18463 a - 21549\bigr] \)
22.1-a5 \( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 1\) , \( -13 a^{3} + 17 a^{2} + 48 a - 41\) , \( -75 a^{3} + 116 a^{2} + 245 a - 240\bigr] \)
22.1-a6 \( \bigl[a^{2} - a - 1\) , \( a^{3} - 2 a^{2} - 3 a + 4\) , \( a^{3} - 4 a - 1\) , \( 2 a^{3} - 3 a^{2} - 7 a + 4\) , \( a^{3} - a^{2} - 4 a\bigr] \)
22.1-a7 \( \bigl[a^{2} - 1\) , \( a^{3} - 3 a - 1\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -29 a^{3} - 37 a^{2} + 29 a + 24\) , \( 247 a^{3} + 345 a^{2} - 177 a - 202\bigr] \)
22.1-a8 \( \bigl[a^{2} - 1\) , \( 0\) , \( a^{3} - 4 a - 1\) , \( -65 a^{3} + 111 a^{2} + 209 a - 254\) , \( -552 a^{3} + 882 a^{2} + 1709 a - 1680\bigr] \)
22.1-a9 \( \bigl[a^{2} - 1\) , \( 0\) , \( a^{3} - 4 a - 1\) , \( 210 a^{3} - 374 a^{2} - 201 a - 214\) , \( 1496 a^{3} - 6224 a^{2} + 4609 a + 4096\bigr] \)
22.1-a10 \( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -115 a^{3} + 187 a^{2} + 362 a - 350\) , \( 926 a^{3} - 1541 a^{2} - 2691 a + 2732\bigr] \)
22.1-a11 \( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -30 a^{3} + 42 a^{2} + 102 a - 90\) , \( -111 a^{3} + 178 a^{2} + 344 a - 362\bigr] \)
22.1-a12 \( \bigl[a + 1\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - a - 1\) , \( -10 a^{3} + 12 a^{2} + 32 a - 15\) , \( 16 a^{3} - 29 a^{2} - 45 a + 56\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrrrrrrrrrr} 1 & 2 & 4 & 12 & 6 & 2 & 4 & 3 & 6 & 12 & 6 & 2 \\ 2 & 1 & 8 & 24 & 12 & 4 & 8 & 6 & 3 & 24 & 12 & 4 \\ 4 & 8 & 1 & 3 & 24 & 8 & 4 & 12 & 24 & 12 & 6 & 2 \\ 12 & 24 & 3 & 1 & 8 & 24 & 12 & 4 & 8 & 4 & 2 & 6 \\ 6 & 12 & 24 & 8 & 1 & 3 & 24 & 2 & 4 & 8 & 4 & 12 \\ 2 & 4 & 8 & 24 & 3 & 1 & 8 & 6 & 12 & 24 & 12 & 4 \\ 4 & 8 & 4 & 12 & 24 & 8 & 1 & 12 & 24 & 3 & 6 & 2 \\ 3 & 6 & 12 & 4 & 2 & 6 & 12 & 1 & 2 & 4 & 2 & 6 \\ 6 & 3 & 24 & 8 & 4 & 12 & 24 & 2 & 1 & 8 & 4 & 12 \\ 12 & 24 & 12 & 4 & 8 & 24 & 3 & 4 & 8 & 1 & 2 & 6 \\ 6 & 12 & 6 & 2 & 4 & 12 & 6 & 2 & 4 & 2 & 1 & 3 \\ 2 & 4 & 2 & 6 & 12 & 4 & 2 & 6 & 12 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph