Properties

Base field 4.4.13025.1
Label 4.4.13025.1-5.1-a
Conductor 5.1
Rank \( 1 \)

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Base field 4.4.13025.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 12 x^{2} + 3 x + 29 \); class number \(1\).

Elliptic curves in class 5.1-a over 4.4.13025.1

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

Curve label Weierstrass Coefficients
5.1-a1 \( \bigl[-\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{25}{4}\) , \( -\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{29}{4}\) , \( a^{2} - a - 5\) , \( -\frac{2735}{4} a^{3} + \frac{3951}{2} a^{2} + \frac{8953}{2} a - \frac{42055}{4}\) , \( -\frac{89057}{4} a^{3} + \frac{128577}{2} a^{2} + \frac{291655}{2} a - \frac{1368261}{4}\bigr] \)
5.1-a2 \( \bigl[-\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{25}{4}\) , \( -\frac{1}{4} a^{3} + \frac{3}{2} a^{2} + \frac{1}{2} a - \frac{29}{4}\) , \( a^{2} - a - 5\) , \( -\frac{15}{2} a^{3} + 23 a^{2} + 44 a - \frac{225}{2}\) , \( \frac{43}{4} a^{3} - \frac{59}{2} a^{2} - \frac{153}{2} a + \frac{671}{4}\bigr] \)
5.1-a3 \( \bigl[1\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a^{2} - \frac{3}{2} a + \frac{25}{4}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{5}{4}\) , \( \frac{61}{2} a^{3} - 103 a^{2} - 113 a + \frac{721}{2}\) , \( -\frac{1013}{4} a^{3} + \frac{1731}{2} a^{2} + \frac{1871}{2} a - \frac{12089}{4}\bigr] \)
5.1-a4 \( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -a - 1\) , \( a^{2} - 5\) , \( -\frac{7}{2} a^{3} + a^{2} + 12 a - \frac{5}{2}\) , \( \frac{39}{4} a^{3} + \frac{27}{2} a^{2} - \frac{65}{2} a - \frac{177}{4}\bigr] \)
5.1-a5 \( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{1}{2} a - \frac{1}{4}\) , \( a^{2} - 5\) , \( -4 a^{2} - 25 a - 29\) , \( -\frac{1}{4} a^{3} - \frac{41}{2} a^{2} - \frac{179}{2} a - \frac{389}{4}\bigr] \)
5.1-a6 \( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{1}{2} a - \frac{1}{4}\) , \( a^{2} - 5\) , \( -\frac{15}{4} a^{3} - \frac{23}{2} a^{2} + \frac{45}{2} a + \frac{249}{4}\) , \( 32 a^{3} - 11 a^{2} - 411 a - 521\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph