Properties

Base field \(\Q(\sqrt{5}, \sqrt{13})\)
Label 4.4.4225.1-1.1-b
Conductor 1.1
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 1.1-b over \(\Q(\sqrt{5}, \sqrt{13})\)

Isogeny class 1.1-b contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
1.1-b1 \( \bigl[\frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{9}{4} a - \frac{5}{2}\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{13}{4} a - \frac{5}{2}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{9}{4} a - \frac{3}{2}\) , \( -\frac{101}{2} a^{3} - 150 a^{2} + \frac{21}{2} a + 59\) , \( -\frac{1937}{2} a^{3} - 2836 a^{2} + \frac{855}{2} a + 1309\bigr] \)
1.1-b2 \( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 2\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 5 a + 2\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( -\frac{29}{2} a^{3} + 7 a^{2} + \frac{255}{2} a - 74\) , \( -\frac{123}{2} a^{3} + 39 a^{2} + \frac{1059}{2} a - 353\bigr] \)
1.1-b3 \( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -\frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{13}{4} a + \frac{7}{2}\) , \( -\frac{1}{4} a^{3} + \frac{11}{4} a + \frac{1}{2}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{9}{4} a + \frac{3}{2}\) , \( \frac{9}{4} a^{3} + \frac{3}{2} a^{2} - \frac{73}{4} a - \frac{25}{2}\bigr] \)
1.1-b4 \( \bigl[-\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{9}{4} a - \frac{1}{2}\) , \( a - 1\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{9}{4} a - \frac{3}{2}\) , \( -\frac{109}{4} a^{3} + \frac{29}{2} a^{2} + \frac{1037}{4} a - \frac{389}{2}\) , \( -205 a^{3} + 109 a^{2} + 1882 a - 1312\bigr] \)
1.1-b5 \( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{3}{2} a - 3\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a + 6\) , \( a^{2} - 2 a - 7\bigr] \)
1.1-b6 \( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 3\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 2\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + 3 a + 4\) , \( -a^{2} + 2 a + 1\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph