Elliptic curve isogeny class 12288.1-b over number field \(\Q(\sqrt{-3}) \)

• Base field: \(\Q(\sqrt{-3}) \)
• Label: 2.0.3.1-12288.1-b
• Conductor: 12288.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 12288.1-b over \(\Q(\sqrt{-3}) \)

Isogeny class 12288.1-b contains 8 curves linked by isogenies of degrees dividing 16.

Curve label	Weierstrass Coefficients
12288.1-b1	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -43 a + 20\) , \( 68 a - 103\bigr] \)
12288.1-b2	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -23 a - 20\) , \( -68 a - 35\bigr] \)
12288.1-b3	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -63 a\) , \( 1377\bigr] \)
12288.1-b4	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -3 a\) , \( -3\bigr] \)
12288.1-b5	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 17 a\) , \( -15\bigr] \)
12288.1-b6	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 97 a\) , \( 385\bigr] \)
12288.1-b7	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 257 a\) , \( -1503\bigr] \)
12288.1-b8	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 1537 a\) , \( 23713\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 16 & 2 & 4 & 8 & 8 & 16 \\ 4 & 1 & 16 & 2 & 4 & 8 & 8 & 16 \\ 16 & 16 & 1 & 8 & 4 & 2 & 8 & 4 \\ 2 & 2 & 8 & 1 & 2 & 4 & 4 & 8 \\ 4 & 4 & 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 8 & 2 & 4 & 2 & 1 & 4 & 2 \\ 8 & 8 & 8 & 4 & 2 & 4 & 1 & 8 \\ 16 & 16 & 4 & 8 & 4 & 2 & 8 & 1 \end{array}\right)\)

Isogeny graph