Properties

Base field \(\Q(\sqrt{5}, \sqrt{21})\)
Label 4.4.11025.1-20.2-b
Conductor 20.2
Rank \( 1 \)

Related objects

Learn more about

Base field \(\Q(\sqrt{5}, \sqrt{21})\)

Generator \(a\), with minimal polynomial \( x^{4} - 13 x^{2} + 16 \); class number \(1\).

Elliptic curves in class 20.2-b over \(\Q(\sqrt{5}, \sqrt{21})\)

Isogeny class 20.2-b contains 4 curves linked by isogenies of degrees dividing 4.

Curve label Weierstrass Coefficients
20.2-b1 \( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{3}{2}\) , \( 114 a^{3} + 137 a^{2} - 1327 a - 1599\) , \( -\frac{1915}{8} a^{3} - \frac{559}{2} a^{2} + \frac{22183}{8} a + \frac{6413}{2}\bigr] \)
20.2-b2 \( \bigl[-\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{1}{2}\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{8} a^{3} + \frac{1}{2} a^{2} + \frac{13}{8} a - \frac{3}{2}\) , \( \frac{667}{8} a^{3} + \frac{199}{2} a^{2} - \frac{7751}{8} a - \frac{2303}{2}\) , \( -\frac{11505}{8} a^{3} - \frac{3379}{2} a^{2} + \frac{133733}{8} a + \frac{39273}{2}\bigr] \)
20.2-b3 \( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{13}{8} a + \frac{7}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{21}{8} a + \frac{9}{2}\) , \( -\frac{1}{2} a^{3} - 3 a^{2} + \frac{11}{2} a + 33\bigr] \)
20.2-b4 \( \bigl[\frac{1}{2} a^{2} + \frac{1}{2} a - 3\) , \( -\frac{1}{8} a^{3} - \frac{1}{2} a^{2} + \frac{13}{8} a + \frac{7}{2}\) , \( -\frac{1}{8} a^{3} + \frac{17}{8} a + \frac{3}{2}\) , \( \frac{1141}{8} a^{3} + \frac{579}{2} a^{2} - \frac{13281}{8} a - \frac{6741}{2}\) , \( -\frac{33289}{8} a^{3} - \frac{12851}{2} a^{2} + \frac{386909}{8} a + \frac{149351}{2}\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph