Elliptic curve isogeny class 512.3-e over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-512.3-e
• Conductor: 512.3
• Rank: \( 0 \)

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

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

Elliptic curves in class 512.3-e over \(\Q(\sqrt{17}) \)

Isogeny class 512.3-e contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
512.3-e1	\( \bigl[0\) , \( a\) , \( 0\) , \( 9 a - 22\) , \( 18 a - 39\bigr] \)
512.3-e2	\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -12 a + 32\) , \( -92 a + 236\bigr] \)
512.3-e3	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -4 a - 8\) , \( 0\bigr] \)
512.3-e4	\( \bigl[0\) , \( -a\) , \( 0\) , \( 94 a - 239\) , \( 655 a - 1678\bigr] \)
512.3-e5	\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -44 a - 128\) , \( -352 a - 416\bigr] \)
512.3-e6	\( \bigl[0\) , \( -a\) , \( 0\) , \( -166 a + 421\) , \( 3275 a - 8378\bigr] \)

Rank

Rank: \( 0 \)

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