Elliptic curve isogeny class 8.4-a over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-8.4-a
• Conductor: 8.4
• Rank: \( 0 \)

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

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

Elliptic curves in class 8.4-a over \(\Q(\sqrt{17}) \)

Isogeny class 8.4-a contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
8.4-a1	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 3 a - 11\) , \( -3 a + 6\bigr] \)
8.4-a2	\( \bigl[a\) , \( 0\) , \( a\) , \( -6 a + 13\) , \( 33 a - 86\bigr] \)
8.4-a3	\( \bigl[a\) , \( a + 1\) , \( a\) , \( 2 a\) , \( a\bigr] \)
8.4-a4	\( \bigl[a\) , \( 0\) , \( a\) , \( -14 a - 23\) , \( -47 a - 74\bigr] \)
8.4-a5	\( \bigl[a\) , \( a + 1\) , \( a\) , \( 7 a - 20\) , \( 15 a - 40\bigr] \)
8.4-a6	\( \bigl[a\) , \( -a\) , \( 0\) , \( -80 a + 205\) , \( -1186 a + 3038\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph