Properties

Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Label 4.4.19600.1-1.1-b
Conductor 1.1
Rank \( 0 \)

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Base field \(\Q(\sqrt{5}, \sqrt{7})\)

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 15 x^{2} + 16 x + 29 \); class number \(1\).

Elliptic curves in class 1.1-b over \(\Q(\sqrt{5}, \sqrt{7})\)

Isogeny class 1.1-b contains 4 curves linked by isogenies of degrees dividing 10.

Curve label Weierstrass Coefficients
1.1-b1 \( \bigl[-\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a + \frac{1}{23}\) , \( \frac{1}{23} a^{3} + \frac{10}{23} a^{2} - \frac{33}{23} a - \frac{81}{23}\) , \( -\frac{1}{23} a^{3} + \frac{13}{23} a^{2} - \frac{13}{23} a - \frac{103}{23}\) , \( \frac{61}{23} a^{3} - \frac{333}{23} a^{2} + \frac{218}{23} a + \frac{579}{23}\) , \( \frac{542}{23} a^{3} - \frac{2837}{23} a^{2} + \frac{1158}{23} a + \frac{4720}{23}\bigr] \)
1.1-b2 \( \bigl[-\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a + \frac{1}{23}\) , \( \frac{1}{23} a^{3} + \frac{10}{23} a^{2} - \frac{33}{23} a - \frac{81}{23}\) , \( -\frac{3}{23} a^{3} + \frac{16}{23} a^{2} + \frac{30}{23} a - \frac{102}{23}\) , \( -\frac{73}{23} a^{3} - \frac{132}{23} a^{2} + \frac{385}{23} a + \frac{370}{23}\) , \( -\frac{544}{23} a^{3} - \frac{1208}{23} a^{2} + \frac{2933}{23} a + \frac{3561}{23}\bigr] \)
1.1-b3 \( \bigl[-\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a + \frac{1}{23}\) , \( \frac{1}{23} a^{3} - \frac{13}{23} a^{2} - \frac{10}{23} a + \frac{80}{23}\) , \( -\frac{1}{23} a^{3} + \frac{13}{23} a^{2} - \frac{13}{23} a - \frac{103}{23}\) , \( \frac{61}{23} a^{3} - \frac{356}{23} a^{2} + \frac{241}{23} a + \frac{740}{23}\) , \( -\frac{470}{23} a^{3} + \frac{2476}{23} a^{2} - \frac{1027}{23} a - \frac{4204}{23}\bigr] \)
1.1-b4 \( \bigl[-\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a + \frac{1}{23}\) , \( \frac{1}{23} a^{3} - \frac{13}{23} a^{2} - \frac{10}{23} a + \frac{80}{23}\) , \( -\frac{3}{23} a^{3} + \frac{16}{23} a^{2} + \frac{30}{23} a - \frac{102}{23}\) , \( -\frac{73}{23} a^{3} - \frac{155}{23} a^{2} + \frac{408}{23} a + \frac{531}{23}\) , \( \frac{468}{23} a^{3} + \frac{1069}{23} a^{2} - \frac{2472}{23} a - \frac{3247}{23}\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph