Properties

Base field 4.4.19225.1
Label 4.4.19225.1-9.2-b
Conductor 9.2
Rank \( 1 \)

Related objects

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Base field 4.4.19225.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 15 x^{2} + 2 x + 44 \); class number \(1\).

Elliptic curves in class 9.2-b over 4.4.19225.1

Isogeny class 9.2-b contains 8 curves linked by isogenies of degrees dividing 12.

Curve label Weierstrass Coefficients
9.2-b1 \( \bigl[-a^{3} + 4 a^{2} + 7 a - 23\) , \( -a^{3} + 4 a^{2} + 5 a - 24\) , \( 0\) , \( \frac{3}{2} a^{3} - \frac{1}{2} a^{2} - \frac{59}{2} a - 40\) , \( 714 a^{3} + 2139 a^{2} - 2226 a - 7767\bigr] \)
9.2-b2 \( \bigl[-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 14\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a^{2} - \frac{5}{2} a + 15\) , \( 1\) , \( -\frac{3}{2} a^{3} - \frac{13}{2} a^{2} + \frac{15}{2} a + 34\) , \( \frac{7}{2} a^{3} + \frac{19}{2} a^{2} - \frac{33}{2} a - 46\bigr] \)
9.2-b3 \( \bigl[-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{7}{2} a - 14\) , \( -a + 1\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 15\) , \( \frac{3}{2} a^{3} - \frac{5}{2} a^{2} - \frac{21}{2} a + 14\) , \( 2 a^{3} - 4 a^{2} - 13 a + 20\bigr] \)
9.2-b4 \( \bigl[-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{7}{2} a - 14\) , \( -a + 1\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 15\) , \( 9 a^{3} - 30 a^{2} - 58 a + 149\) , \( \frac{59}{2} a^{3} - \frac{225}{2} a^{2} - \frac{377}{2} a + 553\bigr] \)
9.2-b5 \( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{7}{2} a + 8\) , \( -\frac{3}{2} a^{3} + \frac{11}{2} a^{2} + \frac{21}{2} a - 32\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 14\) , \( -\frac{217}{2} a^{3} + \frac{763}{2} a^{2} + \frac{1335}{2} a - 1896\) , \( 1600 a^{3} - 5621 a^{2} - 9873 a + 28008\bigr] \)
9.2-b6 \( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{7}{2} a + 8\) , \( a^{3} - 4 a^{2} - 5 a + 23\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{7}{2} a + 9\) , \( -\frac{5}{2} a^{3} + \frac{17}{2} a^{2} + \frac{33}{2} a - 42\) , \( a^{3} - 3 a^{2} - 10 a + 4\bigr] \)
9.2-b7 \( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 9\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 8\) , \( -a^{3} + 4 a^{2} + 6 a - 22\) , \( -\frac{163}{2} a^{3} + \frac{555}{2} a^{2} + \frac{1183}{2} a - 1558\) , \( 641 a^{3} - 2126 a^{2} - 4613 a + 12014\bigr] \)
9.2-b8 \( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 9\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 8\) , \( -a^{3} + 4 a^{2} + 6 a - 22\) , \( -64 a^{3} + 220 a^{2} + 464 a - 1238\) , \( \frac{1809}{2} a^{3} - \frac{6007}{2} a^{2} - \frac{13031}{2} a + 16959\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 12 & 2 & 4 & 4 & 3 & 6 & 12 \\ 12 & 1 & 6 & 12 & 3 & 4 & 2 & 4 \\ 2 & 6 & 1 & 2 & 2 & 6 & 3 & 6 \\ 4 & 12 & 2 & 1 & 4 & 12 & 6 & 3 \\ 4 & 3 & 2 & 4 & 1 & 12 & 6 & 12 \\ 3 & 4 & 6 & 12 & 12 & 1 & 2 & 4 \\ 6 & 2 & 3 & 6 & 6 & 2 & 1 & 2 \\ 12 & 4 & 6 & 3 & 12 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph