Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-31.1-b
Conductor 31.1
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 31.1-b over \(\Q(\zeta_{15})^+\)

Isogeny class 31.1-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
31.1-b1 \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -80 a^{3} + 335 a^{2} + 315 a - 1284\) , \( -2177 a^{3} + 4753 a^{2} + 8378 a - 18117\bigr] \)
31.1-b2 \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -5 a^{3} + 20 a^{2} + 20 a - 79\) , \( -42 a^{3} + 85 a^{2} + 161 a - 317\bigr] \)
31.1-b3 \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -10 a^{3} + 25 a^{2} + 45 a - 74\) , \( 9 a^{3} + 101 a^{2} - 28 a - 357\bigr] \)
31.1-b4 \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( 5 a^{3} - 20 a^{2} - 20 a - 9\) , \( -4 a^{3} + 99 a^{2} + 23 a - 5\bigr] \)
31.1-b5 \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( -4\) , \( -a^{3} + 4 a^{2} + 4 a - 7\bigr] \)
31.1-b6 \( \bigl[a^{2} - 2\) , \( -a^{2} + 1\) , \( 0\) , \( 1\) , \( 0\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

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

Isogeny graph