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

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-256.1-e
• 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-e over \(\Q(\sqrt{17}) \)

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

Curve label	Weierstrass Coefficients
256.1-e1	\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( a + 4\) , \( 0\bigr] \)
256.1-e2	\( \bigl[0\) , \( 1\) , \( 0\) , \( 16 a - 8\) , \( -48 a - 12\bigr] \)
256.1-e3	\( \bigl[0\) , \( 1\) , \( 0\) , \( -4 a - 8\) , \( -8 a - 12\bigr] \)
256.1-e4	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 25 a - 60\) , \( 24 a - 60\bigr] \)
256.1-e5	\( \bigl[0\) , \( -1\) , \( 0\) , \( a - 4\) , \( -22 a + 56\bigr] \)
256.1-e6	\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 305 a - 780\) , \( 4096 a - 10492\bigr] \)

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