Properties

Base field 3.3.316.1
Label 3.3.316.1-8.2-a
Conductor 8.2
Rank \( 0 \)

Related objects

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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 8.2-a over 3.3.316.1

Isogeny class 8.2-a contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
8.2-a1 \( \bigl[a\) , \( -a - 1\) , \( a^{2} + a - 2\) , \( 2190935255 a^{2} - 1159698357 a - 9309591809\) , \( -81630791102268 a^{2} + 43208531179462 a + 346860703632228\bigr] \)
8.2-a2 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -272 a^{2} + 144 a + 1157\) , \( 2826 a^{2} - 1496 a - 12008\bigr] \)
8.2-a3 \( \bigl[a\) , \( a + 1\) , \( a^{2} + a - 2\) , \( -91 a^{2} + 363 a - 351\) , \( -1638 a^{2} + 5248 a - 3310\bigr] \)
8.2-a4 \( \bigl[a\) , \( a + 1\) , \( a^{2} + a - 2\) , \( -6 a^{2} + 23 a - 21\) , \( -25 a^{2} + 80 a - 56\bigr] \)
8.2-a5 \( \bigl[a\) , \( a + 1\) , \( a^{2} + a - 2\) , \( -a^{2} + 3 a - 11\) , \( -72 a^{2} + 200 a - 106\bigr] \)
8.2-a6 \( \bigl[a\) , \( a + 1\) , \( a^{2} + a - 2\) , \( -a^{2} + 3 a - 1\) , \( -2 a\bigr] \)
8.2-a7 \( \bigl[a\) , \( -a^{2} + a + 3\) , \( a\) , \( 14246 a^{2} - 7541 a - 60535\) , \( -1307660 a^{2} + 692166 a + 5556430\bigr] \)
8.2-a8 \( \bigl[a\) , \( -a^{2} + a + 3\) , \( a\) , \( -2419 a^{2} + 1279 a + 10275\) , \( -4236156 a^{2} + 2242268 a + 18000022\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 8 & 16 & 8 & 16 & 4 & 2 & 4 \\ 8 & 1 & 8 & 4 & 8 & 2 & 4 & 8 \\ 16 & 8 & 1 & 2 & 4 & 4 & 8 & 16 \\ 8 & 4 & 2 & 1 & 2 & 2 & 4 & 8 \\ 16 & 8 & 4 & 2 & 1 & 4 & 8 & 16 \\ 4 & 2 & 4 & 2 & 4 & 1 & 2 & 4 \\ 2 & 4 & 8 & 4 & 8 & 2 & 1 & 2 \\ 4 & 8 & 16 & 8 & 16 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph