Elliptic curve isogeny class 9.1-a over number field \(\Q(\sqrt{3}) \)

• Base field: \(\Q(\sqrt{3}) \)
• Label: 2.2.12.1-9.1-a
• Conductor: 9.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 9.1-a over \(\Q(\sqrt{3}) \)

Isogeny class 9.1-a contains 6 curves linked by isogenies of degrees dividing 18.

Curve label	Weierstrass Coefficients
9.1-a1	\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 25 a - 45\) , \( 72 a - 127\bigr] \)
9.1-a2	\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 25 a - 45\) , \( -117 a + 202\bigr] \)
9.1-a3	\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 0\) , \( 0\bigr] \)
9.1-a4	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -7 a + 13\) , \( 6 a - 10\bigr] \)
9.1-a5	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 88 a - 152\) , \( 564 a - 977\bigr] \)
9.1-a6	\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 88 a - 152\) , \( -717 a + 1242\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 9 & 3 & 6 & 2 & 18 \\ 9 & 1 & 3 & 6 & 18 & 2 \\ 3 & 3 & 1 & 2 & 6 & 6 \\ 6 & 6 & 2 & 1 & 3 & 3 \\ 2 & 18 & 6 & 3 & 1 & 9 \\ 18 & 2 & 6 & 3 & 9 & 1 \end{array}\right)\)

Isogeny graph