Properties

Base field \(\Q(\zeta_{15})^+\)
Label 4.4.1125.1-31.4-b
Conductor 31.4
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 31.4-b over \(\Q(\zeta_{15})^+\)

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

Curve label Weierstrass Coefficients
31.4-b1 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( 20 a^{3} + 20 a^{2} - 55 a - 94\) , \( -92 a^{3} - 99 a^{2} + 265 a + 402\bigr] \)
31.4-b2 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -4\) , \( -4 a^{3} - 4 a^{2} + 11 a + 10\bigr] \)
31.4-b3 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( 1\) , \( 0\bigr] \)
31.4-b4 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -20 a^{3} - 20 a^{2} + 55 a + 6\) , \( -92 a^{3} - 85 a^{2} + 241 a + 58\bigr] \)
31.4-b5 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -20 a^{3} - 25 a^{2} + 45 a + 41\) , \( -93 a^{3} - 101 a^{2} + 280 a + 46\bigr] \)
31.4-b6 \( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( a^{3} + a^{2} - 3 a - 3\) , \( 0\) , \( -340 a^{3} - 335 a^{2} + 945 a + 131\) , \( -5083 a^{3} - 4753 a^{2} + 13402 a + 2742\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph