Properties

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

Related objects

Learn more about

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-b over 4.4.2777.1

Isogeny class 22.1-b contains 8 curves linked by isogenies of degrees dividing 20.

Curve label Weierstrass Coefficients
22.1-b1 \( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 574 a^{3} - 1057 a^{2} - 549 a - 385\) , \( 14427 a^{3} - 29786 a^{2} - 11937 a + 4219\bigr] \)
22.1-b2 \( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 9 a^{3} + 8 a^{2} - 59 a - 35\) , \( -37 a^{3} + 17 a^{2} + 136 a + 71\bigr] \)
22.1-b3 \( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{2} - 2\) , \( 4 a^{3} - 7 a^{2} - 9 a + 5\) , \( -a^{3} + 2 a^{2} + a - 3\bigr] \)
22.1-b4 \( \bigl[a^{3} - 4 a - 1\) , \( -a^{3} + 2 a^{2} + 3 a - 4\) , \( a^{3} - a^{2} - 3 a + 2\) , \( -25 a^{3} + 26 a^{2} + 84 a - 75\) , \( -255 a^{3} - 134 a^{2} + 413 a - 111\bigr] \)
22.1-b5 \( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 2 a - 3\) , \( a^{3} - 3 a - 1\) , \( -2 a - 2\) , \( 28 a^{3} - 47 a^{2} - 81 a + 81\bigr] \)
22.1-b6 \( \bigl[a^{3} - 3 a - 1\) , \( a^{2} - 2 a - 3\) , \( a^{2} - a - 1\) , \( -833 a^{3} - 1088 a^{2} + 652 a + 590\) , \( -39688 a^{3} - 53892 a^{2} + 31054 a + 33219\bigr] \)
22.1-b7 \( \bigl[a^{2} - a - 1\) , \( a^{3} - 4 a - 2\) , \( 0\) , \( 4 a^{3} - 15 a^{2} + 7 a + 9\) , \( 26 a^{3} - 63 a^{2} - 10 a + 33\bigr] \)
22.1-b8 \( \bigl[a + 1\) , \( -a^{2} + a + 2\) , \( a\) , \( -506 a^{3} - 392 a^{2} + 539 a - 67\) , \( -15172 a^{3} - 18204 a^{2} + 13108 a + 9055\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 5 & 10 & 4 & 20 & 2 & 20 & 4 \\ 5 & 1 & 2 & 20 & 4 & 10 & 4 & 20 \\ 10 & 2 & 1 & 10 & 2 & 5 & 2 & 10 \\ 4 & 20 & 10 & 1 & 5 & 2 & 20 & 4 \\ 20 & 4 & 2 & 5 & 1 & 10 & 4 & 20 \\ 2 & 10 & 5 & 2 & 10 & 1 & 10 & 2 \\ 20 & 4 & 2 & 20 & 4 & 10 & 1 & 5 \\ 4 & 20 & 10 & 4 & 20 & 2 & 5 & 1 \end{array}\right)\)

Isogeny graph