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

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-612.1-e
• Conductor: 612.1
• Rank: \( 0 \)

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

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

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

Isogeny class 612.1-e contains 8 curves linked by isogenies of degrees dividing 16.

Curve label	Weierstrass Coefficients
612.1-e1	\( \bigl[1\) , \( 0\) , \( 0\) , \( -1644\) , \( -30942\bigr] \)
612.1-e2	\( \bigl[1\) , \( 1\) , \( 1\) , \( 86 a + 131\) , \( 2522 a + 3939\bigr] \)
612.1-e3	\( \bigl[1\) , \( 0\) , \( 0\) , \( 226\) , \( -2232\bigr] \)
612.1-e4	\( \bigl[1\) , \( 0\) , \( 0\) , \( -114\) , \( -396\bigr] \)
612.1-e5	\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( 68\bigr] \)
612.1-e6	\( \bigl[1\) , \( 0\) , \( 0\) , \( -1734\) , \( -27936\bigr] \)
612.1-e7	\( \bigl[1\) , \( 1\) , \( 1\) , \( -86 a + 217\) , \( -2522 a + 6461\bigr] \)
612.1-e8	\( \bigl[1\) , \( 0\) , \( 0\) , \( -27744\) , \( -1781010\bigr] \)

Rank

Rank: \( 0 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 8 & 4 & 8 & 2 & 16 & 4 \\ 16 & 1 & 8 & 4 & 2 & 8 & 4 & 16 \\ 8 & 8 & 1 & 2 & 4 & 4 & 8 & 8 \\ 4 & 4 & 2 & 1 & 2 & 2 & 4 & 4 \\ 8 & 2 & 4 & 2 & 1 & 4 & 2 & 8 \\ 2 & 8 & 4 & 2 & 4 & 1 & 8 & 2 \\ 16 & 4 & 8 & 4 & 2 & 8 & 1 & 16 \\ 4 & 16 & 8 & 4 & 8 & 2 & 16 & 1 \end{array}\right)\)

Isogeny graph