Elliptic curve isogeny class 68.1-a over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-68.1-a
• Conductor: 68.1
• Rank: \( 0 \)

Base field \(\Q(\sqrt{17}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).

Elliptic curves in class 68.1-a over \(\Q(\sqrt{17}) \)

Isogeny class 68.1-a contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
68.1-a1	\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -611 a - 955\) , \( -16227 a - 25345\bigr] \)
68.1-a2	\( \bigl[1\) , \( 1\) , \( 1\) , \( 50 a - 128\) , \( 290 a - 743\bigr] \)
68.1-a3	\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \)
68.1-a4	\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \)
68.1-a5	\( \bigl[1\) , \( 1\) , \( 1\) , \( -50 a - 78\) , \( -290 a - 453\bigr] \)
68.1-a6	\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \)
68.1-a7	\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \)
68.1-a8	\( \bigl[1\) , \( a\) , \( a\) , \( 610 a - 1565\) , \( 16226 a - 41571\bigr] \)

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