Elliptic curve isogeny class 28.2-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-28.2-a
• Conductor: 28.2
• Rank: \( 0 \)

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

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

Elliptic curves in class 28.2-a over \(\Q(\sqrt{2}) \)

Isogeny class 28.2-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
28.2-a1	\( \bigl[0\) , \( -1\) , \( 0\) , \( -18 a - 37\) , \( -68 a - 108\bigr] \)
28.2-a2	\( \bigl[0\) , \( -1\) , \( 0\) , \( 2 a + 3\) , \( 0\bigr] \)
28.2-a3	\( \bigl[a\) , \( -1\) , \( a\) , \( -2 a - 4\) , \( a + 1\bigr] \)
28.2-a4	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 9 a - 18\) , \( 320 a - 467\bigr] \)
28.2-a5	\( \bigl[a\) , \( -1\) , \( a\) , \( -17 a - 29\) , \( -49 a - 66\bigr] \)
28.2-a6	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -a + 2\) , \( -12 a + 17\bigr] \)
28.2-a7	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 34 a - 53\) , \( 133 a - 203\bigr] \)
28.2-a8	\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 19 a - 148\) , \( -318 a - 201\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 6 & 4 & 12 & 12 & 2 & 4 \\ 3 & 1 & 2 & 12 & 4 & 4 & 6 & 12 \\ 6 & 2 & 1 & 6 & 2 & 2 & 3 & 6 \\ 4 & 12 & 6 & 1 & 12 & 3 & 2 & 4 \\ 12 & 4 & 2 & 12 & 1 & 4 & 6 & 3 \\ 12 & 4 & 2 & 3 & 4 & 1 & 6 & 12 \\ 2 & 6 & 3 & 2 & 6 & 6 & 1 & 2 \\ 4 & 12 & 6 & 4 & 3 & 12 & 2 & 1 \end{array}\right)\)

Isogeny graph