Properties

Base field 3.3.316.1
Label 3.3.316.1-8.1-a
Conductor 8.1
Rank \( 0 \)

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Base field 3.3.316.1

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

Elliptic curves in class 8.1-a over 3.3.316.1

Isogeny class 8.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
8.1-a1 \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 4\) , \( a^{2} - 2\) , \( -a^{2} - 8 a + 8\) , \( 5 a^{2} - 22 a + 11\bigr] \)
8.1-a2 \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 4\) , \( a^{2} - 2\) , \( -126 a^{2} - 98 a + 78\) , \( -1365 a^{2} - 1026 a + 787\bigr] \)
8.1-a3 \( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 696542 a^{2} - 368715 a - 2959749\) , \( 463057095 a^{2} - 245103850 a - 1967594769\bigr] \)
8.1-a4 \( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 43522 a^{2} - 23060 a - 184974\) , \( 7250072 a^{2} - 3837642 a - 30806682\bigr] \)
8.1-a5 \( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 8597 a^{2} - 4555 a - 36539\) , \( 638631 a^{2} - 338020 a - 2713601\bigr] \)
8.1-a6 \( \bigl[a\) , \( -a^{2} - a + 2\) , \( a\) , \( 587 a^{2} - 310 a - 2494\) , \( 8248 a^{2} - 4366 a - 35048\bigr] \)
8.1-a7 \( \bigl[a\) , \( -a^{2} - a + 4\) , \( a\) , \( 420286700 a^{2} - 222464805 a - 1785857409\) , \( 6903318103765 a^{2} - 3654040730844 a - 29333168803431\bigr] \)
8.1-a8 \( \bigl[a\) , \( -a^{2} - a + 4\) , \( a\) , \( -15155975 a^{2} + 8022315 a + 64399881\) , \( 50371899485 a^{2} - 26662681574 a - 214037280091\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph