Elliptic curve isogeny class 784.1-e over number field \(\Q(\sqrt{2}) \)

• Base field: \(\Q(\sqrt{2}) \)
• Label: 2.2.8.1-784.1-e
• Conductor: 784.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 784.1-e over \(\Q(\sqrt{2}) \)

Isogeny class 784.1-e contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
784.1-e1	\( \bigl[a\) , \( -a\) , \( 0\) , \( -194 a - 314\) , \( 2200 a + 3186\bigr] \)
784.1-e2	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 6 a + 7\) , \( 23 a + 32\bigr] \)
784.1-e3	\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -15 a + 18\) , \( 30 a - 43\bigr] \)
784.1-e4	\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 5 a - 7\) , \( 7 a - 9\bigr] \)
784.1-e5	\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 25 a - 52\) , \( -84 a + 159\bigr] \)
784.1-e6	\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -115 a - 122\) , \( 322 a + 1587\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrr} 1 & 8 & 8 & 4 & 2 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 8 & 4 & 1 & 2 & 4 & 8 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 2 & 4 & 4 & 2 & 1 & 2 \\ 4 & 8 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph