Properties

Base field 4.4.4525.1
Label 4.4.4525.1-45.3-c
Conductor 45.3
Rank \( 0 \)

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Base field 4.4.4525.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} + 3 x + 9 \); class number \(1\).

Elliptic curves in class 45.3-c over 4.4.4525.1

Isogeny class 45.3-c contains 4 curves linked by isogenies of degrees dividing 6.

Curve label Weierstrass Coefficients
45.3-c1 \( \bigl[\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{7}{3} a - 4\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - 1\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{3} a + 1\) , \( \frac{7}{3} a^{3} - \frac{22}{3} a^{2} + \frac{2}{3} a + 14\) , \( -6 a^{3} + 18 a^{2} + 6 a - 24\bigr] \)
45.3-c2 \( \bigl[\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{7}{3} a - 4\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - 1\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{3} a + 1\) , \( \frac{77}{3} a^{3} - \frac{257}{3} a^{2} - \frac{38}{3} a + 124\) , \( \frac{544}{3} a^{3} - \frac{1693}{3} a^{2} - \frac{349}{3} a + 800\bigr] \)
45.3-c3 \( \bigl[\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{7}{3} a - 3\) , \( -\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - 1\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a\) , \( -\frac{37}{3} a^{3} - \frac{641}{3} a^{2} + \frac{274}{3} a + 344\) , \( -1000 a^{3} - 113 a^{2} + 1747 a + 723\bigr] \)
45.3-c4 \( \bigl[a^{2} - 3\) , \( -\frac{1}{3} a^{3} - \frac{2}{3} a^{2} + \frac{7}{3} a + 5\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{7}{3} a - 4\) , \( -\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - \frac{8}{3} a - 6\) , \( 3 a^{3} - 4 a^{2} - 37 a - 33\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 6 & 3 \\ 2 & 1 & 3 & 6 \\ 6 & 3 & 1 & 2 \\ 3 & 6 & 2 & 1 \end{array}\right)\)

Isogeny graph