Elliptic curve isogeny class 729.1-d over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-729.1-d
• Conductor: 729.1
• Rank: \( 1 \)

Base field \(\Q(\sqrt{2}) \)

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

Elliptic curves in class 729.1-d over \(\Q(\sqrt{2}) \)

Isogeny class 729.1-d contains 4 curves linked by isogenies of degrees dividing 27.

Curve label	Weierstrass Coefficients
729.1-d1	\( \bigl[0\) , \( 0\) , \( 1\) , \( -30\) , \( 63\bigr] \)
729.1-d2	\( \bigl[0\) , \( 0\) , \( 1\) , \( -270\) , \( -1708\bigr] \)
729.1-d3	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 0\bigr] \)
729.1-d4	\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -7\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 27 & 3 & 9 \\ 27 & 1 & 9 & 3 \\ 3 & 9 & 1 & 3 \\ 9 & 3 & 3 & 1 \end{array}\right)\)

Isogeny graph