Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-23.3-a
Conductor 23.3
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 23.3-a over \(\Q(\zeta_{11})^+\)

Isogeny class 23.3-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
23.3-a1 \( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{3} + a^{2} - 4 a - 1\) , \( a^{4} - 4 a^{2} + a + 3\) , \( -32 a^{4} + 731 a^{3} - 356 a^{2} - 1659 a - 761\) , \( -179 a^{4} + 13981 a^{3} - 10713 a^{2} - 28326 a - 8289\bigr] \)
23.3-a2 \( \bigl[a^{4} - 3 a^{2}\) , \( -a^{4} + a^{3} + 4 a^{2} - 4 a - 2\) , \( a^{4} - 3 a^{2} + a + 1\) , \( 18 a^{4} - 30 a^{3} - 46 a^{2} + 82 a - 17\) , \( -37 a^{4} + 76 a^{3} + 93 a^{2} - 205 a + 49\bigr] \)
23.3-a3 \( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{3} + a^{2} - 4 a - 1\) , \( a^{4} - 4 a^{2} + a + 3\) , \( -2 a^{4} + a^{3} + 9 a^{2} - 4 a - 6\) , \( -2 a^{4} + 8 a^{2} - 4\bigr] \)
23.3-a4 \( \bigl[a^{4} - 4 a^{2} + a + 3\) , \( a^{3} + a^{2} - 4 a - 1\) , \( a^{4} - 4 a^{2} + a + 3\) , \( -372 a^{4} + 226 a^{3} + 1434 a^{2} - 609 a - 1191\) , \( -5104 a^{4} + 2964 a^{3} + 20110 a^{2} - 7895 a - 16315\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

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

Isogeny graph