Properties

Base field 5.5.36497.1
Label 5.5.36497.1-39.1-a
Conductor 39.1
Rank not recorded

Related objects

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Base field 5.5.36497.1

Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).

Elliptic curves in class 39.1-a over 5.5.36497.1

Isogeny class 39.1-a contains 8 curves linked by isogenies of degrees dividing 28.

Curve label Weierstrass Coefficients
39.1-a1 \( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 7 a - 3\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( -10 a^{4} - 442 a^{3} + 162 a^{2} + 997 a - 537\) , \( 1117 a^{4} + 6433 a^{3} - 3401 a^{2} - 13458 a + 5199\bigr] \)
39.1-a2 \( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 7 a - 3\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( 25 a^{4} - 77 a^{3} - 28 a^{2} + 177 a - 82\) , \( 195 a^{4} - 507 a^{3} - 353 a^{2} + 1180 a - 360\bigr] \)
39.1-a3 \( \bigl[a^{2} - 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 5 a\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 5 a + 3\) , \( 4 a^{4} - 3 a^{3} - 15 a^{2} + 1\bigr] \)
39.1-a4 \( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 7 a - 3\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( -5 a^{4} - 2 a^{3} + 22 a^{2} + 2 a - 17\) , \( 10 a^{4} - 36 a^{3} - 41 a^{2} + 75 a + 25\bigr] \)
39.1-a5 \( \bigl[a^{2} - 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 5 a\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( 21 a^{4} + 48 a^{3} - 214 a^{2} - 40 a + 28\) , \( -319 a^{4} + 696 a^{3} + 155 a^{2} - 141 a + 20\bigr] \)
39.1-a6 \( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 7 a - 3\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( 2185 a^{4} - 6512 a^{3} - 2423 a^{2} + 15352 a - 7692\) , \( -148316 a^{4} + 370555 a^{3} + 276975 a^{2} - 864563 a + 250675\bigr] \)
39.1-a7 \( \bigl[a^{4} - 2 a^{3} - 2 a^{2} + 4 a\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 7 a - 3\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( -8160 a^{4} + 9948 a^{3} + 35372 a^{2} - 20898 a - 32917\) , \( -915354 a^{4} + 1590551 a^{3} + 3078970 a^{2} - 3551457 a - 1619171\bigr] \)
39.1-a8 \( \bigl[1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( 2 a^{4} - 3 a^{3} - 7 a^{2} + 7 a + 3\) , \( -4706 a^{4} + 2553 a^{3} + 11325 a^{2} - 4556 a - 5334\) , \( 186623 a^{4} + 106339 a^{3} - 305029 a^{2} - 211344 a - 18618\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 28 & 7 & 14 & 28 & 2 & 4 & 4 \\ 28 & 1 & 4 & 2 & 4 & 14 & 7 & 28 \\ 7 & 4 & 1 & 2 & 4 & 14 & 28 & 28 \\ 14 & 2 & 2 & 1 & 2 & 7 & 14 & 14 \\ 28 & 4 & 4 & 2 & 1 & 14 & 28 & 7 \\ 2 & 14 & 14 & 7 & 14 & 1 & 2 & 2 \\ 4 & 7 & 28 & 14 & 28 & 2 & 1 & 4 \\ 4 & 28 & 28 & 14 & 7 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph