Elliptic curve isogeny class 112.2-a over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-112.2-a
• Conductor: 112.2
• Rank: \( 0 \)

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

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

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

Elliptic curves in class 112.2-a over \(\Q(\sqrt{2}) \)

Isogeny class 112.2-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
112.2-a1	\( \bigl[0\) , \( 1\) , \( 0\) , \( -18 a - 37\) , \( 68 a + 108\bigr] \)
112.2-a2	\( \bigl[0\) , \( 1\) , \( 0\) , \( 2 a + 3\) , \( 0\bigr] \)
112.2-a3	\( \bigl[a\) , \( 0\) , \( a\) , \( -2 a - 4\) , \( -a - 2\bigr] \)
112.2-a4	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 7 a - 19\) , \( -312 a + 448\bigr] \)
112.2-a5	\( \bigl[a\) , \( 0\) , \( a\) , \( -17 a - 29\) , \( 49 a + 65\bigr] \)
112.2-a6	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -3 a + 1\) , \( 10 a - 16\bigr] \)
112.2-a7	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 32 a - 54\) , \( -100 a + 149\bigr] \)
112.2-a8	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 17 a - 149\) , \( 336 a + 52\bigr] \)