Properties

Base field 4.4.18625.1
Label 4.4.18625.1-5.1-b
Conductor 5.1
Rank \( 1 \)

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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-b over 4.4.18625.1

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

Curve label Weierstrass Coefficients
5.1-b1 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{4}{3} a - \frac{41}{6}\) , \( \frac{1}{6} a^{3} + a^{2} - \frac{7}{3} a - \frac{47}{6}\) , \( \frac{1}{6} a^{3} - \frac{4}{3} a + \frac{7}{6}\) , \( \frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{5}{6}\) , \( \frac{5}{6} a^{3} + 2 a^{2} - \frac{17}{3} a - \frac{85}{6}\bigr] \)
5.1-b2 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{41}{6}\) , \( a - 1\) , \( a + 1\) , \( \frac{13}{2} a^{3} + 5 a^{2} - 35 a - \frac{85}{2}\) , \( \frac{11}{2} a^{3} + 60 a^{2} - 76 a - \frac{527}{2}\bigr] \)
5.1-b3 \( \bigl[a^{2} - 6\) , \( -a^{2} - a + 6\) , \( a^{2} - 7\) , \( \frac{41}{3} a^{3} + 2 a^{2} - \frac{418}{3} a - \frac{148}{3}\) , \( \frac{245}{6} a^{3} + 63 a^{2} - \frac{1274}{3} a - \frac{4267}{6}\bigr] \)
5.1-b4 \( \bigl[a^{2} - 6\) , \( -a^{2} - a + 6\) , \( a^{2} - 7\) , \( -\frac{1}{2} a^{3} + 2 a^{2} + 4 a - \frac{47}{2}\) , \( -4 a^{3} + 11 a^{2} + 38 a - 106\bigr] \)
5.1-b5 \( \bigl[1\) , \( -\frac{1}{6} a^{3} + a^{2} + \frac{1}{3} a - \frac{49}{6}\) , \( \frac{1}{6} a^{3} - \frac{4}{3} a + \frac{7}{6}\) , \( -\frac{67}{3} a^{3} + 19 a^{2} + \frac{269}{3} a + \frac{8}{3}\) , \( \frac{2762}{3} a^{3} - 3256 a^{2} - \frac{5077}{3} a + \frac{30503}{3}\bigr] \)
5.1-b6 \( \bigl[1\) , \( -\frac{1}{6} a^{3} + a^{2} + \frac{1}{3} a - \frac{49}{6}\) , \( \frac{1}{6} a^{3} - \frac{4}{3} a + \frac{7}{6}\) , \( \frac{37}{2} a^{3} - 81 a^{2} - 22 a + \frac{547}{2}\) , \( 261 a^{3} - 1057 a^{2} - 369 a + 3424\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph