Elliptic curve isogeny class 33.1-c over number field \(\Q(\sqrt{3}) \)

• Base field: \(\Q(\sqrt{3}) \)
• Label: 2.2.12.1-33.1-c
• Conductor: 33.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 33.1-c over \(\Q(\sqrt{3}) \)

Isogeny class 33.1-c contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
33.1-c1	\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -729 a + 1261\) , \( -20243 a + 35060\bigr] \)
33.1-c2	\( \bigl[a\) , \( -1\) , \( 1\) , \( 3362 a - 5824\) , \( -177111 a + 306765\bigr] \)
33.1-c3	\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 20 a - 49\) , \( -100 a + 184\bigr] \)
33.1-c4	\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 296 a - 514\) , \( -3378 a + 5849\bigr] \)
33.1-c5	\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 316 a - 549\) , \( -2833 a + 4905\bigr] \)
33.1-c6	\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 1681 a - 2919\) , \( 48497 a - 84006\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph