Properties

Base field \(\Q(\sqrt{5}, \sqrt{13})\)
Label 4.4.4225.1-16.3-a
Conductor 16.3
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 16.3-a over \(\Q(\sqrt{5}, \sqrt{13})\)

Isogeny class 16.3-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
16.3-a1 \( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{13}{4} a - \frac{7}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( -\frac{669}{4} a^{3} - 487 a^{2} + \frac{291}{4} a + \frac{449}{2}\) , \( \frac{11555}{2} a^{3} + 16878 a^{2} - \frac{5421}{2} a - 7915\bigr] \)
16.3-a2 \( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{5}{4} a - \frac{3}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( -\frac{3}{4} a^{3} + \frac{3}{2} a^{2} - \frac{5}{4} a - \frac{1}{2}\) , \( -\frac{5}{4} a^{3} + 2 a^{2} + \frac{3}{4} a - \frac{3}{2}\bigr] \)
16.3-a3 \( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{1}{4} a^{3} + \frac{11}{4} a + \frac{1}{2}\) , \( a\) , \( \frac{13}{4} a^{3} + \frac{15}{2} a^{2} - \frac{117}{4} a - \frac{117}{2}\) , \( \frac{37}{4} a^{3} + 14 a^{2} - \frac{315}{4} a - \frac{241}{2}\bigr] \)
16.3-a4 \( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{1}{4} a^{3} + \frac{11}{4} a + \frac{1}{2}\) , \( a\) , \( \frac{9}{2} a^{3} + 5 a^{2} - \frac{81}{2} a - 36\) , \( 23 a^{3} + \frac{29}{2} a^{2} - \frac{393}{2} a - 124\bigr] \)
16.3-a5 \( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 2\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( -29 a^{3} - 85 a^{2} + 14 a + 39\) , \( \frac{807}{2} a^{3} + 1179 a^{2} - \frac{375}{2} a - 552\bigr] \)
16.3-a6 \( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + 4 a\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 1\) , \( a^{2} + 2 a + 1\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} + a\bigr] \)
16.3-a7 \( \bigl[a\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{13}{4} a - \frac{3}{2}\) , \( a\) , \( \frac{83}{4} a^{3} + 17 a^{2} - \frac{725}{4} a - \frac{273}{2}\) , \( \frac{403}{4} a^{3} + 72 a^{2} - \frac{3457}{4} a - \frac{1205}{2}\bigr] \)
16.3-a8 \( \bigl[a\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{13}{4} a - \frac{3}{2}\) , \( a\) , \( \frac{13}{4} a^{3} + \frac{9}{2} a^{2} - \frac{125}{4} a - \frac{63}{2}\) , \( -18 a^{3} - \frac{19}{2} a^{2} + \frac{301}{2} a + 88\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph