Properties

Base field \(\Q(\sqrt{2}, \sqrt{5})\)
Label 4.4.1600.1-31.3-a
Conductor 31.3
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 31.3-a over \(\Q(\sqrt{2}, \sqrt{5})\)

Isogeny class 31.3-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
31.3-a1 \( \bigl[\frac{1}{2} a^{3} - 2 a + 1\) , \( \frac{1}{2} a^{3} - a\) , \( \frac{1}{2} a^{3} - 2 a + 1\) , \( -21 a^{3} - \frac{45}{2} a^{2} + 51 a - 22\) , \( -\frac{3}{2} a^{3} + \frac{209}{2} a^{2} + 145 a - 213\bigr] \)
31.3-a2 \( \bigl[\frac{1}{2} a^{3} - 2 a + 1\) , \( \frac{1}{2} a^{3} - a\) , \( \frac{1}{2} a^{3} - 2 a + 1\) , \( -6 a^{3} - 10 a^{2} + 6 a - 2\) , \( -\frac{65}{2} a^{3} - \frac{151}{2} a^{2} + 21 a + 53\bigr] \)
31.3-a3 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a\) , \( -\frac{1}{2} a^{3} + 3 a + 1\) , \( \frac{1}{2} a^{2} + a - 1\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 3 a + 4\) , \( \frac{1}{2} a^{3} + 2 a^{2} + 2 a\bigr] \)
31.3-a4 \( \bigl[\frac{1}{2} a^{2} + a - 1\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - a + 2\) , \( \frac{1}{2} a^{3} - 2 a + 1\) , \( 45 a^{3} - \frac{199}{2} a^{2} - 37 a + 75\) , \( \frac{673}{2} a^{3} - \frac{1537}{2} a^{2} - 256 a + 582\bigr] \)
31.3-a5 \( \bigl[\frac{1}{2} a^{3} - 2 a + 1\) , \( \frac{1}{2} a^{3} - a\) , \( \frac{1}{2} a^{3} - 2 a + 1\) , \( -a^{3} + \frac{5}{2} a^{2} + 6 a - 7\) , \( -\frac{1}{2} a^{3} - \frac{5}{2} a^{2} - a + 3\bigr] \)
31.3-a6 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - a - 1\) , \( a - 1\) , \( \frac{1}{2} a^{2} + a\) , \( 4 a^{3} + 5 a^{2} - 23 a - 29\) , \( -\frac{5}{2} a^{3} - \frac{3}{2} a^{2} + 12 a + 5\bigr] \)
31.3-a7 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - a - 1\) , \( a - 1\) , \( \frac{1}{2} a^{2} + a\) , \( 34 a^{3} + \frac{85}{2} a^{2} - 238 a - 359\) , \( -586 a^{3} - 312 a^{2} + 3301 a + 2167\bigr] \)
31.3-a8 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - 2 a\) , \( a\) , \( \frac{1}{2} a^{3} - a\) , \( 26 a^{3} - \frac{93}{2} a^{2} - 67 a + 69\) , \( -107 a^{3} + 262 a^{2} - 22 a - 121\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph