Elliptic curve isogeny class 49.2-a over number field \(\Q(\sqrt{2}, \sqrt{3})\)

• Base field: \(\Q(\sqrt{2}, \sqrt{3})\)
• Label: 4.4.2304.1-49.2-a
• Conductor: 49.2
• Rank: \( 0 \)

Base field \(\Q(\sqrt{2}, \sqrt{3})\)

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

Elliptic curves in class 49.2-a over \(\Q(\sqrt{2}, \sqrt{3})\)

Isogeny class 49.2-a contains 4 curves linked by isogenies of degrees dividing 10.

Curve label	Weierstrass Coefficients
49.2-a1	\( \bigl[a^{2} - 1\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 4 a\) , \( -a + 1\) , \( a^{3} + a^{2} - 4 a - 3\bigr] \)
49.2-a2	\( \bigl[a^{2} + a - 2\) , \( a^{2} - 1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( 5 a^{3} + 3 a^{2} - 17 a - 6\) , \( 5 a^{3} + 3 a^{2} - 16 a - 9\bigr] \)
49.2-a3	\( \bigl[a^{3} + a^{2} - 4 a - 2\) , \( -a^{3} - a^{2} + 5 a + 3\) , \( 1\) , \( 19 a^{3} - 75 a^{2} + a + 14\) , \( -216 a^{3} + 556 a^{2} + 49 a - 144\bigr] \)
49.2-a4	\( \bigl[a^{2} - 1\) , \( -a^{3} - a^{2} + 5 a + 1\) , \( a\) , \( -182 a^{3} + 80 a^{2} + 706 a - 359\) , \( 1931 a^{3} - 907 a^{2} - 7432 a + 3823\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph