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

• Base field: \(\Q(\sqrt{3}) \)
• Label: 2.2.12.1-512.1-b
• 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-b over \(\Q(\sqrt{3}) \)

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

Curve label	Weierstrass Coefficients
512.1-b1	\( \bigl[0\) , \( -a\) , \( 0\) , \( 220 a - 380\) , \( 2464 a - 4268\bigr] \)
512.1-b2	\( \bigl[0\) , \( -a\) , \( 0\) , \( 15 a - 25\) , \( 38 a - 66\bigr] \)
512.1-b3	\( \bigl[0\) , \( -a\) , \( 0\) , \( -6 a - 11\) , \( 15 a + 26\bigr] \)
512.1-b4	\( \bigl[0\) , \( -a\) , \( 0\) , \( -40 a + 70\) , \( 172 a - 298\bigr] \)

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