Elliptic curve isogeny class 27648.2-e over number field \(\Q(\sqrt{-2}) \)

• Base field: \(\Q(\sqrt{-2}) \)
• Label: 2.0.8.1-27648.2-e
• Conductor: 27648.2
• Rank: \( 2 \)

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

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

Rank

Rank: \( 2 \)

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

Elliptic curves in class 27648.2-e over \(\Q(\sqrt{-2}) \)

Isogeny class 27648.2-e contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
27648.2-e1	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 34 a + 127\) , \( 425 a - 289\bigr] \)
27648.2-e2	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 54 a + 27\) , \( 81 a + 243\bigr] \)
27648.2-e3	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a + 7\) , \( 11 a - 1\bigr] \)
27648.2-e4	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -6 a - 6\) , \( -12 a + 6\bigr] \)