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

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

Isogeny class 612.1-l contains 8 curves linked by isogenies of degrees dividing 12.

Curve label	Weierstrass Coefficients
612.1-l1	\( \bigl[1\) , \( 0\) , \( 1\) , \( -216\) , \( 2062\bigr] \)
612.1-l2	\( \bigl[1\) , \( 0\) , \( 1\) , \( 1809\) , \( -37790\bigr] \)
612.1-l3	\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -1941 a - 3037\) , \( 36391 a + 56829\bigr] \)
612.1-l4	\( \bigl[1\) , \( 0\) , \( 1\) , \( -751\) , \( -6046\bigr] \)
612.1-l5	\( \bigl[1\) , \( 1\) , \( 0\) , \( -10090 a - 15825\) , \( 751560 a + 1173349\bigr] \)
612.1-l6	\( \bigl[1\) , \( 0\) , \( 1\) , \( -256\) , \( 1550\bigr] \)
612.1-l7	\( \bigl[1\) , \( 1\) , \( 0\) , \( 10090 a - 25915\) , \( -751560 a + 1924909\bigr] \)
612.1-l8	\( \bigl[1\) , \( a\) , \( a + 1\) , \( 1940 a - 4978\) , \( -36392 a + 93220\bigr] \)

Rank

Rank: \( 1 \)

Isogeny matrix

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 6 & 12 & 2 & 12 & 4 \\ 3 & 1 & 12 & 2 & 4 & 6 & 4 & 12 \\ 4 & 12 & 1 & 6 & 12 & 2 & 3 & 4 \\ 6 & 2 & 6 & 1 & 2 & 3 & 2 & 6 \\ 12 & 4 & 12 & 2 & 1 & 6 & 4 & 3 \\ 2 & 6 & 2 & 3 & 6 & 1 & 6 & 2 \\ 12 & 4 & 3 & 2 & 4 & 6 & 1 & 12 \\ 4 & 12 & 4 & 6 & 3 & 2 & 12 & 1 \end{array}\right)\)

Isogeny graph