Elliptic curve isogeny class 29.2-d over number field \(\Q(\sqrt{5}, \sqrt{13})\)

• Base field: \(\Q(\sqrt{5}, \sqrt{13})\)
• Label: 4.4.4225.1-29.2-d
• Conductor: 29.2
• Rank: \( 1 \)

Base field \(\Q(\sqrt{5}, \sqrt{13})\)

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

Elliptic curves in class 29.2-d over \(\Q(\sqrt{5}, \sqrt{13})\)

Isogeny class 29.2-d contains 2 curves linked by isogenies of degree 3.

Curve label	Weierstrass Coefficients
29.2-d1	\( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( \frac{1}{4} a^{3} + \frac{1}{2} a^{2} - \frac{13}{4} a - \frac{7}{2}\) , \( \frac{1}{4} a^{3} - \frac{7}{4} a + \frac{1}{2}\) , \( \frac{9}{2} a^{3} + \frac{11}{2} a^{2} - 44 a - 31\) , \( -\frac{53}{4} a^{3} - \frac{13}{2} a^{2} + \frac{429}{4} a + \frac{143}{2}\bigr] \)
29.2-d2	\( \bigl[\frac{1}{2} a^{2} - \frac{1}{2} a - 1\) , \( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + 5 a + 2\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a^{2} + \frac{9}{4} a - \frac{3}{2}\) , \( \frac{19}{2} a^{3} - 8 a^{2} - \frac{161}{2} a + 64\) , \( 43 a^{3} - 30 a^{2} - 367 a + 254\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph