Properties

Base field 4.4.12400.1
Label 4.4.12400.1-9.1-b
Conductor 9.1
Rank \( 1 \)

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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-b over 4.4.12400.1

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

Curve label Weierstrass Coefficients
9.1-b1 \( \bigl[a\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{9}{2} a + \frac{5}{2}\) , \( 0\) , \( -8 a^{3} + 7 a^{2} + 64 a - 78\) , \( \frac{79}{2} a^{3} - 72 a^{2} - \frac{631}{2} a + 604\bigr] \)
9.1-b2 \( \bigl[\frac{1}{2} a^{3} - \frac{7}{2} a\) , \( -\frac{1}{2} a^{2} - a + \frac{7}{2}\) , \( 1\) , \( -\frac{5}{2} a^{3} + \frac{9}{2} a^{2} + \frac{41}{2} a - \frac{71}{2}\) , \( \frac{7}{2} a^{3} - 7 a^{2} - \frac{57}{2} a + 56\bigr] \)
9.1-b3 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{13}{2} a^{3} - a^{2} - \frac{75}{2} a - 15\) , \( -\frac{21}{2} a^{3} + 10 a^{2} + \frac{129}{2} a + 9\bigr] \)
9.1-b4 \( \bigl[\frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{5}{2} a - \frac{7}{2}\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{1}{2} a^{3} - \frac{7}{2} a + 1\) , \( \frac{3}{2} a^{3} + \frac{3}{2} a^{2} - \frac{15}{2} a - \frac{5}{2}\) , \( a^{3} + \frac{5}{2} a^{2} - 2 a - \frac{15}{2}\bigr] \)

Rank

Rank: \( 1 \)

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