Elliptic curve isogeny class 1024.1-k over number field \(\Q(\sqrt{3}) \)

• Base field: \(\Q(\sqrt{3}) \)
• Label: 2.2.12.1-1024.1-k
• Conductor: 1024.1
• Rank: \( 1 \)

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

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

Elliptic curves in class 1024.1-k over \(\Q(\sqrt{3}) \)

Isogeny class 1024.1-k contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
1024.1-k1	\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( a - 2\bigr] \)
1024.1-k2	\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( -a + 2\bigr] \)
1024.1-k3	\( \bigl[0\) , \( -a\) , \( 0\) , \( 170 a - 299\) , \( 1711 a - 2958\bigr] \)
1024.1-k4	\( \bigl[0\) , \( a\) , \( 0\) , \( 170 a - 299\) , \( -1711 a + 2958\bigr] \)
1024.1-k5	\( \bigl[0\) , \( -a\) , \( 0\) , \( 10 a - 19\) , \( 31 a - 54\bigr] \)
1024.1-k6	\( \bigl[0\) , \( a\) , \( 0\) , \( 10 a - 19\) , \( -31 a + 54\bigr] \)
1024.1-k7	\( \bigl[0\) , \( -a\) , \( 0\) , \( 10 a - 59\) , \( -49 a + 2\bigr] \)
1024.1-k8	\( \bigl[0\) , \( a\) , \( 0\) , \( 10 a - 59\) , \( 49 a - 2\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 12 & 2 & 6 & 4 & 12 \\ 3 & 1 & 12 & 4 & 6 & 2 & 12 & 4 \\ 4 & 12 & 1 & 12 & 2 & 6 & 4 & 3 \\ 12 & 4 & 12 & 1 & 6 & 2 & 3 & 4 \\ 2 & 6 & 2 & 6 & 1 & 3 & 2 & 6 \\ 6 & 2 & 6 & 2 & 3 & 1 & 6 & 2 \\ 4 & 12 & 4 & 3 & 2 & 6 & 1 & 12 \\ 12 & 4 & 3 & 4 & 6 & 2 & 12 & 1 \end{array}\right)\)

Isogeny graph