Elliptic curve isogeny class 3072.1-by over number field \(\Q(\sqrt{3}) \)

• Base field: \(\Q(\sqrt{3}) \)
• Label: 2.2.12.1-3072.1-by
• Conductor: 3072.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 3072.1-by over \(\Q(\sqrt{3}) \)

Isogeny class 3072.1-by contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
3072.1-by1	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4 a - 7\) , \( 134 a + 232\bigr] \)
3072.1-by2	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -504 a - 872\) , \( -2120 a - 3672\bigr] \)
3072.1-by3	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -284 a - 492\) , \( 3672 a + 6360\bigr] \)
3072.1-by4	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -4544 a - 7872\) , \( 223656 a + 387384\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

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

Isogeny graph