Elliptic curve isogeny class 648.3-d over number field \(\Q(\sqrt{17}) \)

• Base field: \(\Q(\sqrt{17}) \)
• Label: 2.2.17.1-648.3-d
• Conductor: 648.3
• Rank: \( 1 \)

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

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

Elliptic curves in class 648.3-d over \(\Q(\sqrt{17}) \)

Isogeny class 648.3-d contains 6 curves linked by isogenies of degrees dividing 8.

Curve label	Weierstrass Coefficients
648.3-d1	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 51 a + 80\) , \( 999 a + 1560\bigr] \)
648.3-d2	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 717 a + 1118\) , \( -29467 a - 46014\bigr] \)
648.3-d3	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( 123 a - 316\) , \( -1017 a + 2604\bigr] \)
648.3-d4	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -3 a - 10\) , \( a - 2\bigr] \)
648.3-d5	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -39 a - 61\) , \( -244 a - 380\bigr] \)
648.3-d6	\( \bigl[a + 1\) , \( 1\) , \( a + 1\) , \( -48 a - 145\) , \( 379 a + 340\bigr] \)

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