Properties

Base field \(\Q(\sqrt{5}, \sqrt{13})\)
Label 4.4.4225.1-9.1-a
Conductor 9.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{5}, \sqrt{13})\)

Generator \(a\), with minimal polynomial \( x^{4} - 9 x^{2} + 4 \); class number \(1\).

Elliptic curves in class 9.1-a over \(\Q(\sqrt{5}, \sqrt{13})\)

Isogeny class 9.1-a contains 8 curves linked by isogenies of degrees dividing 16.

Curve label Weierstrass Coefficients
9.1-a1 \( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 3\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{883}{2} a^{3} + 299 a^{2} + \frac{7557}{2} a - 2587\) , \( -\frac{46471}{4} a^{3} + 7963 a^{2} + \frac{396341}{4} a - \frac{135709}{2}\bigr] \)
9.1-a2 \( \bigl[\frac{1}{4} a^{3} - \frac{7}{4} a - \frac{1}{2}\) , \( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - 4 a + 1\) , \( -\frac{1}{4} a^{3} + \frac{11}{4} a + \frac{3}{2}\) , \( -\frac{3}{4} a^{3} + \frac{3}{2} a^{2} + \frac{7}{4} a + \frac{3}{2}\) , \( 6 a^{3} - 17 a^{2} - 5 a + 7\bigr] \)
9.1-a3 \( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -a - 1\) , \( 0\) , \( -a^{3} + 7 a + 6\) , \( \frac{3}{4} a^{3} + 2 a^{2} - \frac{41}{4} a - \frac{13}{2}\bigr] \)
9.1-a4 \( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -a - 1\) , \( 0\) , \( \frac{1}{4} a^{3} - \frac{7}{4} a - \frac{3}{2}\) , \( 0\bigr] \)
9.1-a5 \( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 3\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{111}{4} a^{3} + 19 a^{2} + \frac{949}{4} a - \frac{329}{2}\) , \( -\frac{319}{2} a^{3} + 109 a^{2} + \frac{2723}{2} a - 933\bigr] \)
9.1-a6 \( \bigl[-\frac{1}{4} a^{3} + \frac{11}{4} a + \frac{3}{2}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{9}{4} a + \frac{3}{2}\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{9}{4} a - \frac{3}{2}\) , \( -\frac{55}{4} a^{3} - 40 a^{2} + \frac{33}{4} a + \frac{45}{2}\) , \( -\frac{513}{4} a^{3} - \frac{751}{2} a^{2} + \frac{233}{4} a + \frac{351}{2}\bigr] \)
9.1-a7 \( \bigl[\frac{1}{4} a^{3} - \frac{7}{4} a + \frac{1}{2}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{5}{4} a - \frac{1}{2}\) , \( a\) , \( 2 a^{3} - a^{2} - 19 a + 13\) , \( -\frac{45}{2} a^{3} + \frac{31}{2} a^{2} + 191 a - 131\bigr] \)
9.1-a8 \( \bigl[-\frac{1}{4} a^{3} + \frac{11}{4} a + \frac{3}{2}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{9}{4} a + \frac{3}{2}\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{9}{4} a - \frac{3}{2}\) , \( -\frac{35}{4} a^{3} - 35 a^{2} - \frac{47}{4} a + \frac{15}{2}\) , \( -\frac{679}{4} a^{3} - 499 a^{2} + \frac{401}{4} a + \frac{497}{2}\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 8 & 4 & 2 & 8 & 4 & 16 \\ 16 & 1 & 8 & 4 & 8 & 2 & 16 & 4 \\ 8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 2 & 8 & 4 & 2 & 1 & 4 & 2 & 8 \\ 8 & 2 & 4 & 2 & 4 & 1 & 8 & 2 \\ 4 & 16 & 8 & 4 & 2 & 8 & 1 & 16 \\ 16 & 4 & 8 & 4 & 8 & 2 & 16 & 1 \end{array}\right)\)

Isogeny graph