Properties

Base field \(\Q(\sqrt{5}, \sqrt{7})\)
Label 4.4.19600.1-1.1-c
Conductor 1.1
Rank not recorded

Related objects

Learn more about

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-c over \(\Q(\sqrt{5}, \sqrt{7})\)

Isogeny class 1.1-c contains 8 curves linked by isogenies of degrees dividing 28.

Curve label Weierstrass Coefficients
1.1-c1 \( \bigl[-\frac{1}{23} a^{3} + \frac{13}{23} a^{2} - \frac{13}{23} a - \frac{103}{23}\) , \( \frac{2}{23} a^{3} - \frac{3}{23} a^{2} - \frac{43}{23} a + \frac{22}{23}\) , \( -\frac{1}{23} a^{3} + \frac{13}{23} a^{2} - \frac{13}{23} a - \frac{103}{23}\) , \( \frac{4}{23} a^{3} - \frac{75}{23} a^{2} - \frac{17}{23} a + \frac{159}{23}\) , \( -\frac{30}{23} a^{3} - \frac{24}{23} a^{2} + \frac{70}{23} a + \frac{130}{23}\bigr] \)
1.1-c2 \( \bigl[-\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a - \frac{22}{23}\) , \( -1\) , \( -\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a - \frac{22}{23}\) , \( -\frac{540}{23} a^{3} + \frac{810}{23} a^{2} + \frac{11610}{23} a - \frac{22454}{23}\) , \( -\frac{6446}{23} a^{3} + \frac{9669}{23} a^{2} + \frac{138589}{23} a - \frac{267073}{23}\bigr] \)
1.1-c3 \( \bigl[-\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a - \frac{22}{23}\) , \( -1\) , \( -\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a - \frac{22}{23}\) , \( \frac{540}{23} a^{3} - \frac{810}{23} a^{2} - \frac{11610}{23} a - \frac{10574}{23}\) , \( \frac{6446}{23} a^{3} - \frac{9669}{23} a^{2} - \frac{138589}{23} a - \frac{125261}{23}\bigr] \)
1.1-c4 \( \bigl[-\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a - \frac{22}{23}\) , \( -1\) , \( -\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a - \frac{22}{23}\) , \( \frac{510}{23} a^{3} - \frac{765}{23} a^{2} - \frac{10965}{23} a - \frac{9984}{23}\) , \( \frac{7338}{23} a^{3} - \frac{11007}{23} a^{2} - \frac{157767}{23} a - \frac{142589}{23}\bigr] \)
1.1-c5 \( \bigl[1\) , \( -1\) , \( 1\) , \( -\frac{540}{23} a^{3} + \frac{810}{23} a^{2} + \frac{11610}{23} a - \frac{22385}{23}\) , \( \frac{6446}{23} a^{3} - \frac{9669}{23} a^{2} - \frac{138589}{23} a + \frac{267027}{23}\bigr] \)
1.1-c6 \( \bigl[1\) , \( -1\) , \( 1\) , \( \frac{540}{23} a^{3} - \frac{810}{23} a^{2} - \frac{11610}{23} a - \frac{10505}{23}\) , \( -\frac{6446}{23} a^{3} + \frac{9669}{23} a^{2} + \frac{138589}{23} a + \frac{125215}{23}\bigr] \)
1.1-c7 \( \bigl[1\) , \( -1\) , \( 1\) , \( \frac{510}{23} a^{3} - \frac{765}{23} a^{2} - \frac{10965}{23} a - \frac{9915}{23}\) , \( -\frac{7338}{23} a^{3} + \frac{11007}{23} a^{2} + \frac{157767}{23} a + \frac{142543}{23}\bigr] \)
1.1-c8 \( \bigl[-\frac{3}{23} a^{3} + \frac{16}{23} a^{2} + \frac{30}{23} a - \frac{102}{23}\) , \( \frac{2}{23} a^{3} - \frac{3}{23} a^{2} - \frac{43}{23} a + \frac{22}{23}\) , \( -\frac{2}{23} a^{3} + \frac{3}{23} a^{2} + \frac{43}{23} a + \frac{1}{23}\) , \( \frac{7}{23} a^{3} - \frac{114}{23} a^{2} - \frac{47}{23} a + \frac{560}{23}\) , \( \frac{105}{23} a^{3} - \frac{261}{23} a^{2} - \frac{268}{23} a + \frac{649}{23}\bigr] \)

Rank

Rank not yet determined.

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 28 & 4 & 2 & 4 & 28 & 14 & 7 \\ 28 & 1 & 7 & 14 & 28 & 4 & 2 & 4 \\ 4 & 7 & 1 & 2 & 4 & 28 & 14 & 28 \\ 2 & 14 & 2 & 1 & 2 & 14 & 7 & 14 \\ 4 & 28 & 4 & 2 & 1 & 7 & 14 & 28 \\ 28 & 4 & 28 & 14 & 7 & 1 & 2 & 4 \\ 14 & 2 & 14 & 7 & 14 & 2 & 1 & 2 \\ 7 & 4 & 28 & 14 & 28 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph