Properties

Base field 4.4.18625.1
Label 4.4.18625.1-5.1-d
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-d over 4.4.18625.1

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

Curve label Weierstrass Coefficients
5.1-d1 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{4}{3} a - \frac{41}{6}\) , \( -\frac{1}{6} a^{3} + \frac{7}{3} a - \frac{7}{6}\) , \( \frac{1}{6} a^{3} + a^{2} - \frac{4}{3} a - \frac{35}{6}\) , \( 8 a^{3} - 4 a^{2} - 34 a - 10\) , \( \frac{83}{6} a^{3} - 64 a^{2} - \frac{38}{3} a + \frac{1283}{6}\bigr] \)
5.1-d2 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{4}{3} a - \frac{41}{6}\) , \( -\frac{1}{6} a^{3} + \frac{7}{3} a - \frac{7}{6}\) , \( \frac{1}{6} a^{3} + a^{2} - \frac{4}{3} a - \frac{35}{6}\) , \( -\frac{7}{6} a^{3} - 4 a^{2} + \frac{28}{3} a + \frac{125}{6}\) , \( -5 a^{3} - 12 a^{2} + 34 a + 65\bigr] \)
5.1-d3 \( \bigl[a^{2} - 6\) , \( -\frac{1}{6} a^{3} + a^{2} + \frac{7}{3} a - \frac{43}{6}\) , \( \frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{41}{6}\) , \( \frac{2}{3} a^{3} + 2 a^{2} - \frac{16}{3} a - \frac{31}{3}\) , \( \frac{2}{3} a^{3} + 3 a^{2} - \frac{13}{3} a - \frac{64}{3}\bigr] \)
5.1-d4 \( \bigl[\frac{1}{6} a^{3} + a^{2} - \frac{1}{3} a - \frac{35}{6}\) , \( 0\) , \( 0\) , \( \frac{23}{2} a^{3} + 22 a^{2} - 93 a - \frac{317}{2}\) , \( \frac{7}{2} a^{3} + 19 a^{2} + 30 a + \frac{29}{2}\bigr] \)
5.1-d5 \( \bigl[1\) , \( -a^{2} + a + 7\) , \( 1\) , \( 53 a^{3} + 81 a^{2} - 552 a - 908\) , \( 690 a^{3} + 1011 a^{2} - 7158 a - 11443\bigr] \)
5.1-d6 \( \bigl[1\) , \( -a^{2} + a + 7\) , \( 1\) , \( -\frac{121}{3} a^{3} - 19 a^{2} + \frac{1229}{3} a + \frac{806}{3}\) , \( \frac{7238}{3} a^{3} + 3325 a^{2} - \frac{74980}{3} a - \frac{113659}{3}\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph