Elliptic curve isogeny class 128.6-d over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-128.6-d
• Conductor: 128.6
• Rank: \( 1 \)

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

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

Elliptic curves in class 128.6-d over \(\Q(\sqrt{17}) \)

Isogeny class 128.6-d contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
128.6-d1	\( \bigl[a\) , \( -a\) , \( 0\) , \( -a\) , \( 197 a + 308\bigr] \)
128.6-d2	\( \bigl[a\) , \( 0\) , \( a\) , \( 31 a - 88\) , \( -142 a + 356\bigr] \)
128.6-d3	\( \bigl[a\) , \( 0\) , \( a\) , \( a - 8\) , \( -2 a + 4\bigr] \)
128.6-d4	\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 83 a - 216\) , \( 136 a - 350\bigr] \)

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