Properties

Base field 4.4.13025.1
Label 4.4.13025.1-19.2-a
Conductor 19.2
Rank \( 0 \)

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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 19.2-a over 4.4.13025.1

Isogeny class 19.2-a contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
19.2-a1 \( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( a^{2} - 6\) , \( a\) , \( \frac{15}{2} a^{3} + 5 a^{2} - 76 a - \frac{215}{2}\) , \( \frac{7}{2} a^{3} + 3 a^{2} - 36 a - \frac{113}{2}\bigr] \)
19.2-a2 \( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( a - 1\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{9}{2} a - \frac{49}{4}\) , \( -\frac{1}{4} a^{3} - \frac{9}{2} a^{2} - \frac{21}{2} a - \frac{25}{4}\bigr] \)
19.2-a3 \( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( a - 1\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a + \frac{9}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{1}{2} a - \frac{9}{4}\) , \( -1\bigr] \)
19.2-a4 \( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{9}{4}\) , \( a^{2} - a - 5\) , \( 0\) , \( -\frac{13}{2} a^{3} + 26 a^{2} + 39 a - \frac{281}{2}\) , \( 28 a^{3} - 61 a^{2} - 199 a + 282\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph