Elliptic curve isogeny class 512.1-e over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-512.1-e
• Conductor: 512.1
• Rank: \( 0 \)

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

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

Elliptic curves in class 512.1-e over \(\Q(\sqrt{17}) \)

Isogeny class 512.1-e contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
512.1-e1	\( \bigl[0\) , \( 1\) , \( 0\) , \( 0\) , \( 8 a + 12\bigr] \)
512.1-e2	\( \bigl[0\) , \( -1\) , \( 0\) , \( -768 a - 1200\) , \( -7764 a - 12124\bigr] \)
512.1-e3	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 71 a - 178\) , \( 444 a - 1136\bigr] \)
512.1-e4	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 106 a - 273\) , \( -87 a + 223\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

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

Isogeny graph