Elliptic curve isogeny class 128.5-d over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-128.5-d
• Conductor: 128.5
• Rank: \( 1 \)

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

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

Elliptic curves in class 128.5-d over \(\Q(\sqrt{17}) \)

Isogeny class 128.5-d contains 4 curves linked by isogenies of degrees dividing 4.

Curve label	Weierstrass Coefficients
128.5-d1	\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( a - 1\) , \( -197 a + 505\bigr] \)
128.5-d2	\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -85 a - 133\) , \( -137 a - 214\bigr] \)
128.5-d3	\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3 a - 7\) , \( a + 2\bigr] \)
128.5-d4	\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -33 a - 57\) , \( 141 a + 214\bigr] \)

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