Elliptic curve isogeny class 256.1-f over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-256.1-f
• Conductor: 256.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 256.1-f over \(\Q(\sqrt{17}) \)

Isogeny class 256.1-f contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
256.1-f1	\( \bigl[0\) , \( -1\) , \( 0\) , \( -a - 3\) , \( 22 a + 34\bigr] \)
256.1-f2	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( a + 4\) , \( 4\bigr] \)
256.1-f3	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -23 a - 36\) , \( 0\bigr] \)
256.1-f4	\( \bigl[0\) , \( 1\) , \( 0\) , \( 4 a - 12\) , \( 8 a - 20\bigr] \)
256.1-f5	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -303 a - 476\) , \( -3792 a - 5920\bigr] \)
256.1-f6	\( \bigl[0\) , \( 1\) , \( 0\) , \( -16 a + 8\) , \( 48 a - 60\bigr] \)

Rank

Rank: \( 1 \)

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