Elliptic curve isogeny class 500.1-d over number field \(\Q(\sqrt{6}) \)

• Base field: \(\Q(\sqrt{6}) \)
• Label: 2.2.24.1-500.1-d
• Conductor: 500.1
• Rank: \( 1 \)

Base field \(\Q(\sqrt{6}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 6 \); class number \(1\).

Elliptic curves in class 500.1-d over \(\Q(\sqrt{6}) \)

Isogeny class 500.1-d contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
500.1-d1	\( \bigl[a\) , \( a\) , \( a\) , \( -153 a - 383\) , \( -10624 a - 26027\bigr] \)
500.1-d2	\( \bigl[a\) , \( a\) , \( 0\) , \( 932 a - 2270\) , \( 23858 a - 58376\bigr] \)
500.1-d3	\( \bigl[0\) , \( a\) , \( 0\) , \( 146 a - 362\) , \( -1644 a + 4023\bigr] \)
500.1-d4	\( \bigl[a\) , \( a\) , \( 0\) , \( 22 a - 80\) , \( 746 a - 1918\bigr] \)
500.1-d5	\( \bigl[a\) , \( a\) , \( 0\) , \( 57 a - 145\) , \( 358 a - 876\bigr] \)
500.1-d6	\( \bigl[a\) , \( a\) , \( a\) , \( -328 a - 808\) , \( -5504 a - 13482\bigr] \)
500.1-d7	\( \bigl[0\) , \( a\) , \( 0\) , \( -834 a - 2042\) , \( 19846 a + 48613\bigr] \)
500.1-d8	\( \bigl[a\) , \( a\) , \( a\) , \( -2673 a + 6547\) , \( -732674 a + 1794673\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph