Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-31.4-a
Conductor 31.4
Rank \( 0 \)

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Base field \(\Q(\zeta_{15})^+\)

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

Elliptic curves in class 31.4-a over \(\Q(\zeta_{15})^+\)

Isogeny class 31.4-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
31.4-a1 \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -71 a^{3} + 341 a^{2} + 296 a - 1252\) , \( -1813 a^{3} + 4283 a^{2} + 7065 a - 16149\bigr] \)
31.4-a2 \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -341 a^{3} + 156 a^{2} + 1091 a - 1022\) , \( 4485 a^{3} - 2644 a^{2} - 14997 a + 14364\bigr] \)
31.4-a3 \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -21 a^{3} + 21 a^{2} + 66 a - 102\) , \( 51 a^{3} + 19 a^{2} - 157 a - 1\bigr] \)
31.4-a4 \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -6 a^{3} + 21 a^{2} + 21 a - 77\) , \( -21 a^{3} + 61 a^{2} + 84 a - 229\bigr] \)
31.4-a5 \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( -a^{3} + a^{2} + a - 2\) , \( -a^{3} + a^{2} + 4 a - 6\bigr] \)
31.4-a6 \( \bigl[a^{2} - 1\) , \( -a^{3} + 3 a - 1\) , \( a^{3} - 2 a\) , \( 59 a^{3} - 114 a^{2} - 239 a + 418\) , \( 45 a^{3} + 134 a^{2} - 121 a - 414\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph