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

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-612.1-g
• 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-g over \(\Q(\sqrt{17}) \)

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

Curve label	Weierstrass Coefficients
612.1-g1	\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 16 a - 62\) , \( 342 a + 192\bigr] \)
612.1-g2	\( \bigl[1\) , \( -a\) , \( a + 1\) , \( 10 a - 9\) , \( -15 a + 12\bigr] \)
612.1-g3	\( \bigl[1\) , \( a + 1\) , \( a\) , \( 75 a - 189\) , \( -255 a + 653\bigr] \)
612.1-g4	\( \bigl[1\) , \( -a - 1\) , \( a\) , \( -24 a - 42\) , \( 130 a + 196\bigr] \)
612.1-g5	\( \bigl[1\) , \( a + 1\) , \( a\) , \( -255 a + 651\) , \( -1815 a + 4637\bigr] \)
612.1-g6	\( \bigl[1\) , \( a - 1\) , \( a\) , \( -3 a - 3\) , \( 111 a - 279\bigr] \)

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