Properties

Base field 4.4.18625.1
Label 4.4.18625.1-5.1-a
Conductor 5.1
Rank not recorded

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Base field 4.4.18625.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 14 x^{2} + 9 x + 41 \); class number \(1\).

Elliptic curves in class 5.1-a over 4.4.18625.1

Isogeny class 5.1-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label Weierstrass Coefficients
5.1-a1 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{4}{3} a - \frac{41}{6}\) , \( a^{2} - 8\) , \( 1\) , \( \frac{7}{6} a^{3} + 2 a^{2} - \frac{25}{3} a - \frac{47}{6}\) , \( \frac{1}{2} a^{3} + 5 a^{2} - 6 a - \frac{47}{2}\bigr] \)
5.1-a2 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{41}{6}\) , \( \frac{1}{6} a^{3} - \frac{4}{3} a + \frac{1}{6}\) , \( \frac{1}{6} a^{3} + a^{2} - \frac{4}{3} a - \frac{41}{6}\) , \( -\frac{23}{3} a^{3} + 59 a^{2} - \frac{41}{3} a - \frac{629}{3}\) , \( 105 a^{3} - 358 a^{2} - 206 a + 1110\bigr] \)
5.1-a3 \( \bigl[a^{2} - 6\) , \( \frac{1}{6} a^{3} + a^{2} - \frac{7}{3} a - \frac{47}{6}\) , \( a\) , \( -\frac{3}{2} a^{3} + 4 a^{2} + 12 a - \frac{47}{2}\) , \( -\frac{7}{6} a^{3} + 6 a^{2} + \frac{31}{3} a - \frac{301}{6}\bigr] \)
5.1-a4 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{35}{6}\) , \( -a^{2} + a + 7\) , \( a\) , \( 9 a^{3} + 18 a^{2} - 73 a - 122\) , \( \frac{163}{6} a^{3} + 50 a^{2} - \frac{673}{3} a - \frac{2273}{6}\bigr] \)
5.1-a5 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{35}{6}\) , \( -a^{2} + a + 7\) , \( a\) , \( -\frac{11}{6} a^{3} + 3 a^{2} + \frac{131}{3} a + \frac{373}{6}\) , \( \frac{119}{2} a^{3} + 105 a^{2} - 525 a - \frac{1759}{2}\bigr] \)
5.1-a6 \( \bigl[\frac{1}{6} a^{3} - \frac{1}{3} a + \frac{1}{6}\) , \( -\frac{1}{6} a^{3} - a^{2} + \frac{1}{3} a + \frac{35}{6}\) , \( \frac{1}{6} a^{3} - \frac{1}{3} a + \frac{1}{6}\) , \( -\frac{5}{6} a^{3} - a^{2} + \frac{50}{3} a - \frac{137}{6}\) , \( -\frac{17}{6} a^{3} - a^{2} + \frac{194}{3} a - \frac{653}{6}\bigr] \)

Rank

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Isogeny matrix

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

Isogeny graph