Elliptic curve isogeny class 612.1-d over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-612.1-d
• Conductor: 612.1
• Rank: \( 1 \)

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

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

Elliptic curves in class 612.1-d over \(\Q(\sqrt{17}) \)

Isogeny class 612.1-d contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
612.1-d1	\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 2 a - 6\) , \( -112 a - 168\bigr] \)
612.1-d2	\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 256 a + 395\) , \( 1559 a + 2427\bigr] \)
612.1-d3	\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -74 a - 115\) , \( 329 a + 513\bigr] \)
612.1-d4	\( \bigl[1\) , \( a + 1\) , \( a\) , \( 25 a - 66\) , \( -106 a + 260\bigr] \)
612.1-d5	\( \bigl[1\) , \( a - 1\) , \( a\) , \( -11 a + 2\) , \( 14 a - 2\bigr] \)
612.1-d6	\( \bigl[1\) , \( a + 1\) , \( a\) , \( -15 a - 46\) , \( -358 a + 488\bigr] \)

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