Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-23.4-a
Conductor 23.4
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 23.4-a over \(\Q(\zeta_{11})^+\)

Isogeny class 23.4-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
23.4-a1 \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( -a^{4} - a^{3} + 5 a^{2} + 3 a - 5\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( -2681 a^{4} - 1331 a^{3} + 7125 a^{2} + 587 a - 4049\) , \( -146736 a^{4} - 108973 a^{3} + 354020 a^{2} + 126719 a - 132130\bigr] \)
23.4-a2 \( \bigl[a^{4} + a^{3} - 3 a^{2} - 2 a\) , \( -a^{4} - a^{3} + 5 a^{2} + 3 a - 5\) , \( a^{4} + a^{3} - 4 a^{2} - 2 a + 3\) , \( -136 a^{4} - 116 a^{3} + 365 a^{2} + 142 a - 269\) , \( -2624 a^{4} - 2311 a^{3} + 6124 a^{2} + 3032 a - 2236\bigr] \)
23.4-a3 \( \bigl[a^{4} + a^{3} - 3 a^{2} - 3 a\) , \( a^{4} + a^{3} - 3 a^{2} - 3 a + 1\) , \( a^{4} + a^{3} - 4 a^{2} - 3 a + 2\) , \( -7 a^{4} + 26 a^{3} + 4 a^{2} - 50 a + 12\) , \( -45 a^{4} + 84 a^{3} + 135 a^{2} - 274 a + 66\bigr] \)
23.4-a4 \( \bigl[a^{3} - 2 a + 1\) , \( a^{4} - 4 a^{2} + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a\) , \( a^{3} - 2 a\bigr] \)

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