Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-31.2-a
Conductor 31.2
Rank not recorded

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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 31.2-a over \(\Q(\zeta_{15})^+\)

Isogeny class 31.2-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
31.2-a1 \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -15 a^{3} - 85 a^{2} + 385 a - 331\) , \( 187 a^{3} - 1626 a^{2} + 3723 a - 2518\bigr] \)
31.2-a2 \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 270 a^{3} - 70 a^{2} - 655 a - 231\) , \( -2943 a^{3} + 1542 a^{2} + 6186 a + 1507\bigr] \)
31.2-a3 \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 15 a^{3} - 5 a^{2} - 25 a - 31\) , \( -47 a^{3} + 4 a^{2} + 161 a - 22\bigr] \)
31.2-a4 \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -5 a^{2} + 20 a - 21\) , \( -21 a^{2} + 62 a - 44\bigr] \)
31.2-a5 \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( -1\) , \( -a^{2} + 2 a - 1\bigr] \)
31.2-a6 \( \bigl[a + 1\) , \( a^{3} - 3 a\) , \( a^{2} - 1\) , \( 60 a^{2} - 115 a + 9\) , \( -59 a^{3} - 14 a^{2} + 312 a - 163\bigr] \)

Rank

Rank not yet determined.

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