Properties

Base field 4.4.12400.1
Label 4.4.12400.1-9.1-a
Conductor 9.1
Rank \( 0 \)

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Base field 4.4.12400.1

Generator \(a\), with minimal polynomial \( x^{4} - 12 x^{2} + 31 \); class number \(1\).

Elliptic curves in class 9.1-a over 4.4.12400.1

Isogeny class 9.1-a contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
9.1-a1 \( \bigl[\frac{1}{2} a^{2} - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a - \frac{9}{2}\) , \( a\) , \( -\frac{37}{2} a^{3} + 35 a^{2} + \frac{301}{2} a - 289\) , \( -102 a^{3} + \frac{403}{2} a^{2} + 846 a - \frac{3309}{2}\bigr] \)
9.1-a2 \( \bigl[\frac{1}{2} a^{2} - \frac{5}{2}\) , \( -\frac{1}{2} a^{3} + \frac{1}{2} a^{2} + \frac{5}{2} a - \frac{9}{2}\) , \( a\) , \( -a^{3} + \frac{5}{2} a^{2} + 8 a - \frac{33}{2}\) , \( -\frac{1}{2} a^{3} + \frac{11}{2} a - 8\bigr] \)
9.1-a3 \( \bigl[\frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} + \frac{9}{2} a\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( 7 a^{3} + \frac{13}{2} a^{2} - 53 a - \frac{143}{2}\) , \( 31 a^{3} + \frac{139}{2} a^{2} - 265 a - \frac{1101}{2}\bigr] \)
9.1-a4 \( \bigl[\frac{1}{2} a^{2} + a - \frac{7}{2}\) , \( -\frac{1}{2} a^{3} + \frac{9}{2} a\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a + 1\) , \( -\frac{1}{2} a^{3} - \frac{7}{2} a^{2} + \frac{9}{2} a + \frac{47}{2}\) , \( \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{9}{2} a - \frac{17}{2}\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph