Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-31.3-a
Conductor 31.3
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 31.3-a over \(\Q(\zeta_{15})^+\)

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

Curve label Weierstrass Coefficients
31.3-a1 \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 495 a^{3} + 69 a^{2} - 1642 a - 353\) , \( -7129 a^{3} - 1543 a^{2} + 24030 a + 5034\bigr] \)
31.3-a2 \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 410 a^{3} + 84 a^{2} - 1572 a - 328\) , \( 6096 a^{3} + 1625 a^{2} - 22572 a - 4736\bigr] \)
31.3-a3 \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 40 a^{3} + 4 a^{2} - 142 a - 28\) , \( -32 a^{3} - 5 a^{2} + 76 a + 16\bigr] \)
31.3-a4 \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 25 a^{3} + 4 a^{2} - 97 a - 18\) , \( 82 a^{3} + 20 a^{2} - 308 a - 64\bigr] \)
31.3-a5 \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -a^{2} - 2 a + 2\) , \( 2 a^{3} - 8 a - 1\bigr] \)
31.3-a6 \( \bigl[a^{3} - 2 a\) , \( a^{3} - a^{2} - 3 a + 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( -175 a^{3} - 61 a^{2} + 638 a + 137\) , \( 89 a^{3} + 13 a^{2} - 402 a - 82\bigr] \)

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