Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-31.3-b
Conductor 31.3
Rank \( 0 \)

Related objects

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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 31.3-b over \(\Q(\zeta_{15})^+\)

Isogeny class 31.3-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
31.3-b1 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a^{3} - 75 a^{2} + 320 a - 384\) , \( 330 a^{3} - 1847 a^{2} + 3763 a - 2740\bigr] \)
31.3-b2 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a^{2} - 20 a - 64\) , \( -7 a^{3} - 11 a^{2} + 120 a + 219\bigr] \)
31.3-b3 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a^{2} + 20 a - 24\) , \( 7 a^{3} - 35 a^{2} + 64 a - 35\bigr] \)
31.3-b4 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -4\) , \( -a^{2} + 4 a + 4\bigr] \)
31.3-b5 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 1\) , \( 0\bigr] \)
31.3-b6 \( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -5 a^{3} - 15 a^{2} + 40 a + 16\) , \( -8 a^{3} + a^{2} + 125 a - 166\bigr] \)

Rank

Rank: \( 0 \)

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