Properties

Base field \(\Q(\sqrt{2}, \sqrt{5})\)
Label 4.4.1600.1-31.1-a
Conductor 31.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{2}, \sqrt{5})\)

Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} + 4 \); class number \(1\).

Elliptic curves in class 31.1-a over \(\Q(\sqrt{2}, \sqrt{5})\)

Isogeny class 31.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
31.1-a1 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a - 1\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 3 a + 2\) , \( \frac{1}{2} a^{3} - a\) , \( 43 a^{3} + \frac{93}{2} a^{2} - 208 a - 211\) , \( -226 a^{3} - \frac{349}{2} a^{2} + 1236 a + 1026\bigr] \)
31.1-a2 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a\) , \( -\frac{1}{2} a^{2} + 2\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a\) , \( \frac{1709}{2} a^{3} - \frac{3885}{2} a^{2} - 671 a + 1460\) , \( -43216 a^{3} + 98857 a^{2} + 33055 a - 75482\bigr] \)
31.1-a3 \( \bigl[1\) , \( \frac{1}{2} a^{2}\) , \( \frac{1}{2} a^{2} + a - 1\) , \( a^{3} - \frac{3}{2} a^{2} - 2 a + 1\) , \( 2 a^{3} - \frac{11}{2} a^{2} - a + 4\bigr] \)
31.1-a4 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a\) , \( -\frac{1}{2} a^{2} + 2\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a\) , \( 72 a^{3} - 165 a^{2} - 56 a + 125\) , \( -162 a^{3} + 370 a^{2} + 124 a - 283\bigr] \)
31.1-a5 \( \bigl[\frac{1}{2} a^{2}\) , \( -a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a - 1\) , \( -\frac{27}{2} a^{3} + 2 a^{2} + 46 a - 59\) , \( -43 a^{3} - \frac{69}{2} a^{2} + 108 a - 83\bigr] \)
31.1-a6 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a - 1\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - a\) , \( \frac{1}{2} a^{3} - 2 a + 1\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( 3 a^{3} - \frac{7}{2} a^{2} - 13 a + 12\bigr] \)
31.1-a7 \( \bigl[1\) , \( \frac{1}{2} a^{2}\) , \( \frac{1}{2} a^{2} + a - 1\) , \( 16 a^{3} - 29 a^{2} - 22 a + 11\) , \( \frac{199}{2} a^{3} - 241 a^{2} - 57 a + 197\bigr] \)
31.1-a8 \( \bigl[\frac{1}{2} a^{3} - 2 a + 1\) , \( -\frac{1}{2} a^{3} + a\) , \( 1\) , \( -38 a^{3} + \frac{49}{2} a^{2} + 185 a - 164\) , \( 125 a^{3} - \frac{275}{2} a^{2} - 686 a + 630\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 12 & 4 & 6 & 3 & 12 & 2 & 4 \\ 12 & 1 & 12 & 2 & 4 & 4 & 6 & 3 \\ 4 & 12 & 1 & 6 & 12 & 3 & 2 & 4 \\ 6 & 2 & 6 & 1 & 2 & 2 & 3 & 6 \\ 3 & 4 & 12 & 2 & 1 & 4 & 6 & 12 \\ 12 & 4 & 3 & 2 & 4 & 1 & 6 & 12 \\ 2 & 6 & 2 & 3 & 6 & 6 & 1 & 2 \\ 4 & 3 & 4 & 6 & 12 & 12 & 2 & 1 \end{array}\right)\)

Isogeny graph