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

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-512.1-f
• Conductor: 512.1
• Rank: \( 1 \)

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

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

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

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

Curve label	Weierstrass Coefficients
512.1-f1	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 5 a - 12\) , \( -50 a + 128\bigr] \)
512.1-f2	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 5 a - 60\) , \( -90 a + 144\bigr] \)
512.1-f3	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -5 a - 10\) , \( 6 a + 16\bigr] \)
512.1-f4	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -90 a - 145\) , \( 633 a + 993\bigr] \)

Rank

Rank: \( 1 \)

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