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

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-512.1-i
• 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-i over \(\Q(\sqrt{17}) \)

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

Curve label	Weierstrass Coefficients
512.1-i1	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 14 a + 24\) , \( 15 a + 24\bigr] \)
512.1-i2	\( \bigl[0\) , \( -a\) , \( 0\) , \( a - 1\) , \( 0\bigr] \)
512.1-i3	\( \bigl[0\) , \( -a\) , \( 0\) , \( 11 a - 31\) , \( -26 a + 66\bigr] \)
512.1-i4	\( \bigl[0\) , \( -a\) , \( 0\) , \( -4 a + 4\) , \( 16\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph