Elliptic curve isogeny class 256.1-c over number field \(\Q(\sqrt{3}) \)

• Base field: \(\Q(\sqrt{3}) \)
• Label: 2.2.12.1-256.1-c
• Conductor: 256.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 256.1-c over \(\Q(\sqrt{3}) \)

Isogeny class 256.1-c contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
256.1-c1	\( \bigl[0\) , \( -a\) , \( 0\) , \( 1\) , \( 0\bigr] \)
256.1-c2	\( \bigl[0\) , \( a\) , \( 0\) , \( 1\) , \( 0\bigr] \)
256.1-c3	\( \bigl[0\) , \( -a\) , \( 0\) , \( 20 a - 44\) , \( 92 a - 160\bigr] \)
256.1-c4	\( \bigl[0\) , \( a\) , \( 0\) , \( 20 a - 44\) , \( -92 a + 160\bigr] \)
256.1-c5	\( \bigl[0\) , \( -a\) , \( 0\) , \( -4\) , \( 4 a\bigr] \)
256.1-c6	\( \bigl[0\) , \( a\) , \( 0\) , \( -4\) , \( -4 a\bigr] \)
256.1-c7	\( \bigl[0\) , \( a\) , \( 0\) , \( -20 a - 44\) , \( -92 a - 160\bigr] \)
256.1-c8	\( \bigl[0\) , \( -a\) , \( 0\) , \( -20 a - 44\) , \( 92 a + 160\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph