Elliptic curve isogeny class 512.1-g over number field \(\Q(\sqrt{3}) \)

• Base field: \(\Q(\sqrt{3}) \)
• Label: 2.2.12.1-512.1-g
• Conductor: 512.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 512.1-g over \(\Q(\sqrt{3}) \)

Isogeny class 512.1-g contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
512.1-g1	\( \bigl[0\) , \( a\) , \( 0\) , \( 3064 a - 5306\) , \( 119716 a - 207354\bigr] \)
512.1-g2	\( \bigl[0\) , \( a\) , \( 0\) , \( 209 a - 361\) , \( 1410 a - 2442\bigr] \)
512.1-g3	\( \bigl[0\) , \( a\) , \( 0\) , \( 6 a - 11\) , \( -15 a + 26\bigr] \)
512.1-g4	\( \bigl[0\) , \( a\) , \( 0\) , \( -556 a + 964\) , \( 10464 a - 18124\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph