Properties

Base field \(\Q(\zeta_{9})^+\)
Label 3.3.81.1-51.1-a
Conductor 51.1
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 51.1-a over \(\Q(\zeta_{9})^+\)

Isogeny class 51.1-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
51.1-a1 \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 417 a^{2} - 133 a - 1203\) , \( 4217 a^{2} - 237 a - 14449\bigr] \)
51.1-a2 \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( 27 a^{2} - 58 a - 43\) , \( 50 a^{2} - 62 a + 9\bigr] \)
51.1-a3 \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -33 a^{2} + 92 a - 33\) , \( -292 a^{2} + 546 a + 23\bigr] \)
51.1-a4 \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 2\) , \( 11 a^{2} - 8 a - 6\bigr] \)
51.1-a5 \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -3 a^{2} + 17 a + 7\) , \( 17 a^{2} - 13 a - 7\bigr] \)
51.1-a6 \( \bigl[a^{2} + a - 1\) , \( a^{2} + a - 1\) , \( 0\) , \( -963 a^{2} + 1517 a + 577\) , \( -24013 a^{2} + 36905 a + 15491\bigr] \)

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