Properties

Base field \(\Q(\zeta_{9})^+\)
Label 3.3.81.1-51.3-a
Conductor 51.3
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 51.3-a over \(\Q(\zeta_{9})^+\)

Isogeny class 51.3-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
51.3-a1 \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -543 a^{2} - 968 a - 272\) , \( -13153 a^{2} - 24560 a - 6934\bigr] \)
51.3-a2 \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( 42 a^{2} + 22 a - 82\) , \( -59 a^{2} + 88 a + 322\bigr] \)
51.3-a3 \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -48 a^{2} - 38 a - 12\) , \( -255 a^{2} - 344 a - 86\bigr] \)
51.3-a4 \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -3 a^{2} - 8 a - 7\) , \( a^{2} + 4 a + 4\bigr] \)
51.3-a5 \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -3 a^{2} - 8 a - 2\) , \( 5 a^{2} + 10 a + 2\bigr] \)
51.3-a6 \( \bigl[a + 1\) , \( -a^{2} + a + 1\) , \( a^{2} - 2\) , \( -273 a^{2} + 412 a + 168\) , \( -3801 a^{2} + 3940 a + 1822\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

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

Isogeny graph