Properties

Base field \(\Q(\zeta_{9})^+\)
Label 3.3.81.1-17.3-a
Conductor 17.3
Rank not recorded

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Base field \(\Q(\zeta_{9})^+\)

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

Elliptic curves in class 17.3-a over \(\Q(\zeta_{9})^+\)

Isogeny class 17.3-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
17.3-a1 \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 28433 a^{2} - 9539 a - 82497\) , \( -3025589 a^{2} + 1055150 a + 8703613\bigr] \)
17.3-a2 \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 1913 a^{2} - 759 a - 5327\) , \( -44365 a^{2} + 15660 a + 127267\bigr] \)
17.3-a3 \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 1773 a^{2} - 589 a - 5152\) , \( -46301 a^{2} + 16115 a + 133250\bigr] \)
17.3-a4 \( \bigl[a^{2} - 2\) , \( -a^{2} - a + 1\) , \( a\) , \( 98 a^{2} - 19 a - 307\) , \( -668 a^{2} + 195 a + 1990\bigr] \)
17.3-a5 \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( 35 a^{2} - 8 a - 113\) , \( 133 a^{2} - 47 a - 411\bigr] \)
17.3-a6 \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( -35 a^{2} + 102 a - 63\) , \( -639 a^{2} + 789 a + 777\bigr] \)
17.3-a7 \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( 7 a - 8\) , \( -7 a^{2} + 11 a + 7\bigr] \)
17.3-a8 \( \bigl[a^{2} + a - 2\) , \( a^{2} + a - 2\) , \( a^{2} + a - 1\) , \( 2 a + 2\) , \( a^{2} + a - 1\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

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

Isogeny graph