Elliptic curve isogeny class 46.4-a over number field \(\Q(\sqrt{2}, \sqrt{3})\)

• Base field: \(\Q(\sqrt{2}, \sqrt{3})\)
• Label: 4.4.2304.1-46.4-a
• Conductor: 46.4
• 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 46.4-a over \(\Q(\sqrt{2}, \sqrt{3})\)

Isogeny class 46.4-a contains 3 curves linked by isogenies of degrees dividing 9.

Curve label	Weierstrass Coefficients
46.4-a1	\( \bigl[1\) , \( a^{2} + a - 3\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 4 a^{3} - 7 a^{2} - 5 a + 3\) , \( -9 a^{3} + 67 a - 35\bigr] \)
46.4-a2	\( \bigl[a^{3} - 4 a\) , \( -a^{3} + 4 a + 1\) , \( a^{3} + a^{2} - 4 a - 1\) , \( -a^{3} + 4 a\) , \( 36 a^{3} - 19 a^{2} - 134 a + 69\bigr] \)
46.4-a3	\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a - 1\) , \( a^{2} - 2\) , \( -45 a^{3} - 27 a^{2} + 195 a + 46\) , \( -213 a^{3} - 83 a^{2} + 679 a + 533\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph