Elliptic curve isogeny class 512.4-e over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-512.4-e
• Conductor: 512.4
• Rank: \( 0 \)

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

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

Elliptic curves in class 512.4-e over \(\Q(\sqrt{17}) \)

Isogeny class 512.4-e contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
512.4-e1	\( \bigl[0\) , \( -a\) , \( 0\) , \( 12 a + 20\) , \( 92 a + 144\bigr] \)
512.4-e2	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 166 a + 255\) , \( -3275 a - 5103\bigr] \)
512.4-e3	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -94 a - 145\) , \( -655 a - 1023\bigr] \)
512.4-e4	\( \bigl[0\) , \( a\) , \( 0\) , \( 4 a - 12\) , \( 0\bigr] \)
512.4-e5	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -9 a - 13\) , \( -18 a - 21\bigr] \)
512.4-e6	\( \bigl[0\) , \( a\) , \( 0\) , \( 44 a - 172\) , \( 352 a - 768\bigr] \)

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