Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-23.2-a
Conductor 23.2
Rank not recorded

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Base field \(\Q(\zeta_{11})^+\)

Generator \(a\), with minimal polynomial \( x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1 \); class number \(1\).

Elliptic curves in class 23.2-a over \(\Q(\zeta_{11})^+\)

Isogeny class 23.2-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
23.2-a1 \( \bigl[a^{4} + a^{3} - 4 a^{2} - 3 a + 3\) , \( a^{4} - a^{3} - 3 a^{2} + 2 a\) , \( a^{4} - 3 a^{2}\) , \( -a^{3} - a^{2} + a\) , \( -2 a^{3} - a^{2} + 4 a\bigr] \)
23.2-a2 \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a + 1\) , \( a^{2} + a - 1\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 2\) , \( 49 a^{4} - 67 a^{3} - 121 a^{2} + 192 a - 44\) , \( 233 a^{4} - 388 a^{3} - 573 a^{2} + 1088 a - 256\bigr] \)
23.2-a3 \( \bigl[a^{3} - 2 a\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( a^{3} + a^{2} - 2 a - 2\) , \( -187 a^{4} - 46 a^{3} + 506 a^{2} - 163 a - 437\) , \( -3375 a^{4} - 1423 a^{3} + 9018 a^{2} - 208 a - 5628\bigr] \)
23.2-a4 \( \bigl[a^{3} + a^{2} - 3 a - 1\) , \( -a^{4} - a^{3} + 5 a^{2} + 2 a - 3\) , \( a^{4} - 4 a^{2} + a + 3\) , \( -4278 a^{4} + 3350 a^{3} + 17601 a^{2} - 8315 a - 15116\) , \( -209580 a^{4} + 154663 a^{3} + 874447 a^{2} - 386208 a - 735950\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph