Properties

Base field \(\Q(\sqrt{5}, \sqrt{6})\)
Label 4.4.14400.1-1.1-a
Conductor 1.1
Rank \( 0 \)

Related objects

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Base field \(\Q(\sqrt{5}, \sqrt{6})\)

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 19 \); class number \(1\).

Elliptic curves in class 1.1-a over \(\Q(\sqrt{5}, \sqrt{6})\)

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

Curve label Weierstrass Coefficients
1.1-a1 \( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( 0\) , \( -\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a\) , \( -\frac{34}{19} a^{3} + \frac{89}{19} a^{2} + \frac{401}{19} a - 47\) , \( -\frac{136}{19} a^{3} + \frac{394}{19} a^{2} + \frac{1395}{19} a - 163\bigr] \)
1.1-a2 \( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( 0\) , \( -\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a\) , \( \frac{36}{19} a^{3} - \frac{16}{19} a^{2} - \frac{514}{19} a - 22\) , \( \frac{138}{19} a^{3} - \frac{17}{19} a^{2} - \frac{1812}{19} a - 75\bigr] \)
1.1-a3 \( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( 0\) , \( -\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a\) , \( \frac{6}{19} a^{3} - \frac{47}{19} a^{2} + \frac{117}{19} a - 9\) , \( \frac{60}{19} a^{3} - \frac{280}{19} a^{2} + \frac{11}{19} a + 26\bigr] \)
1.1-a4 \( \bigl[-\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a - 1\) , \( 0\) , \( -\frac{2}{19} a^{3} + \frac{3}{19} a^{2} + \frac{37}{19} a\) , \( -\frac{4}{19} a^{3} - \frac{32}{19} a^{2} - \frac{78}{19} a - 4\) , \( -\frac{58}{19} a^{3} - \frac{103}{19} a^{2} + \frac{332}{19} a + 16\bigr] \)

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